:: ZF_LANG semantic presentation
:: deftheorem Def1 defines VAR ZF_LANG:def 1 :
:: deftheorem Def2 defines x. ZF_LANG:def 2 :
for b
1 being
Nat holds
x. b
1 = 5
+ b
1;
:: deftheorem Def3 defines '=' ZF_LANG:def 3 :
:: deftheorem Def4 defines 'in' ZF_LANG:def 4 :
theorem Th1: :: ZF_LANG:1
canceled;
theorem Th2: :: ZF_LANG:2
canceled;
theorem Th3: :: ZF_LANG:3
canceled;
theorem Th4: :: ZF_LANG:4
canceled;
theorem Th5: :: ZF_LANG:5
canceled;
theorem Th6: :: ZF_LANG:6
for b
1, b
2, b
3, b
4 being
Variable holds
( b
1 '=' b
2 = b
3 '=' b
4 implies ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th7: :: ZF_LANG:7
for b
1, b
2, b
3, b
4 being
Variable holds
( b
1 'in' b
2 = b
3 'in' b
4 implies ( b
1 = b
3 & b
2 = b
4 ) )
:: deftheorem Def5 defines 'not' ZF_LANG:def 5 :
:: deftheorem Def6 defines '&' ZF_LANG:def 6 :
theorem Th8: :: ZF_LANG:8
canceled;
theorem Th9: :: ZF_LANG:9
canceled;
theorem Th10: :: ZF_LANG:10
:: deftheorem Def7 defines All ZF_LANG:def 7 :
theorem Th11: :: ZF_LANG:11
canceled;
theorem Th12: :: ZF_LANG:12
definition
func WFF -> non
empty set means :
Def8:
:: ZF_LANG:def 8
( ( for b
1 being
set holds
( b
1 in a
1 implies b
1 is
FinSequence of
NAT ) ) & ( for b
1, b
2 being
Variable holds
( b
1 '=' b
2 in a
1 & b
1 'in' b
2 in a
1 ) ) & ( for b
1 being
FinSequence of
NAT holds
( b
1 in a
1 implies
'not' b
1 in a
1 ) ) & ( for b
1, b
2 being
FinSequence of
NAT holds
( b
1 in a
1 & b
2 in a
1 implies b
1 '&' b
2 in a
1 ) ) & ( for b
1 being
Variablefor b
2 being
FinSequence of
NAT holds
( b
2 in a
1 implies
All b
1,b
2 in a
1 ) ) & ( for b
1 being non
empty set holds
( ( for b
2 being
set holds
( b
2 in b
1 implies b
2 is
FinSequence of
NAT ) ) & ( for b
2, b
3 being
Variable holds
( b
2 '=' b
3 in b
1 & b
2 'in' b
3 in b
1 ) ) & ( for b
2 being
FinSequence of
NAT holds
( b
2 in b
1 implies
'not' b
2 in b
1 ) ) & ( for b
2, b
3 being
FinSequence of
NAT holds
( b
2 in b
1 & b
3 in b
1 implies b
2 '&' b
3 in b
1 ) ) & ( for b
2 being
Variablefor b
3 being
FinSequence of
NAT holds
( b
3 in b
1 implies
All b
2,b
3 in b
1 ) ) implies a
1 c= b
1 ) ) );
existence
ex b1 being non empty set st
( ( for b2 being set holds
( b2 in b1 implies b2 is FinSequence of NAT ) ) & ( for b2, b3 being Variable holds
( b2 '=' b3 in b1 & b2 'in' b3 in b1 ) ) & ( for b2 being FinSequence of NAT holds
( b2 in b1 implies 'not' b2 in b1 ) ) & ( for b2, b3 being FinSequence of NAT holds
( b2 in b1 & b3 in b1 implies b2 '&' b3 in b1 ) ) & ( for b2 being Variable
for b3 being FinSequence of NAT holds
( b3 in b1 implies All b2,b3 in b1 ) ) & ( for b2 being non empty set holds
( ( for b3 being set holds
( b3 in b2 implies b3 is FinSequence of NAT ) ) & ( for b3, b4 being Variable holds
( b3 '=' b4 in b2 & b3 'in' b4 in b2 ) ) & ( for b3 being FinSequence of NAT holds
( b3 in b2 implies 'not' b3 in b2 ) ) & ( for b3, b4 being FinSequence of NAT holds
( b3 in b2 & b4 in b2 implies b3 '&' b4 in b2 ) ) & ( for b3 being Variable
for b4 being FinSequence of NAT holds
( b4 in b2 implies All b3,b4 in b2 ) ) implies b1 c= b2 ) ) )
uniqueness
for b1, b2 being non empty set holds
( ( for b3 being set holds
( b3 in b1 implies b3 is FinSequence of NAT ) ) & ( for b3, b4 being Variable holds
( b3 '=' b4 in b1 & b3 'in' b4 in b1 ) ) & ( for b3 being FinSequence of NAT holds
( b3 in b1 implies 'not' b3 in b1 ) ) & ( for b3, b4 being FinSequence of NAT holds
( b3 in b1 & b4 in b1 implies b3 '&' b4 in b1 ) ) & ( for b3 being Variable
for b4 being FinSequence of NAT holds
( b4 in b1 implies All b3,b4 in b1 ) ) & ( for b3 being non empty set holds
( ( for b4 being set holds
( b4 in b3 implies b4 is FinSequence of NAT ) ) & ( for b4, b5 being Variable holds
( b4 '=' b5 in b3 & b4 'in' b5 in b3 ) ) & ( for b4 being FinSequence of NAT holds
( b4 in b3 implies 'not' b4 in b3 ) ) & ( for b4, b5 being FinSequence of NAT holds
( b4 in b3 & b5 in b3 implies b4 '&' b5 in b3 ) ) & ( for b4 being Variable
for b5 being FinSequence of NAT holds
( b5 in b3 implies All b4,b5 in b3 ) ) implies b1 c= b3 ) ) & ( for b3 being set holds
( b3 in b2 implies b3 is FinSequence of NAT ) ) & ( for b3, b4 being Variable holds
( b3 '=' b4 in b2 & b3 'in' b4 in b2 ) ) & ( for b3 being FinSequence of NAT holds
( b3 in b2 implies 'not' b3 in b2 ) ) & ( for b3, b4 being FinSequence of NAT holds
( b3 in b2 & b4 in b2 implies b3 '&' b4 in b2 ) ) & ( for b3 being Variable
for b4 being FinSequence of NAT holds
( b4 in b2 implies All b3,b4 in b2 ) ) & ( for b3 being non empty set holds
( ( for b4 being set holds
( b4 in b3 implies b4 is FinSequence of NAT ) ) & ( for b4, b5 being Variable holds
( b4 '=' b5 in b3 & b4 'in' b5 in b3 ) ) & ( for b4 being FinSequence of NAT holds
( b4 in b3 implies 'not' b4 in b3 ) ) & ( for b4, b5 being FinSequence of NAT holds
( b4 in b3 & b5 in b3 implies b4 '&' b5 in b3 ) ) & ( for b4 being Variable
for b5 being FinSequence of NAT holds
( b5 in b3 implies All b4,b5 in b3 ) ) implies b2 c= b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def8 defines WFF ZF_LANG:def 8 :
:: deftheorem Def9 defines ZF-formula-like ZF_LANG:def 9 :
theorem Th13: :: ZF_LANG:13
canceled;
theorem Th14: :: ZF_LANG:14
:: deftheorem Def10 defines being_equality ZF_LANG:def 10 :
:: deftheorem Def11 defines being_membership ZF_LANG:def 11 :
:: deftheorem Def12 defines negative ZF_LANG:def 12 :
:: deftheorem Def13 defines conjunctive ZF_LANG:def 13 :
:: deftheorem Def14 defines universal ZF_LANG:def 14 :
theorem Th15: :: ZF_LANG:15
canceled;
theorem Th16: :: ZF_LANG:16
for b
1 being
ZF-formula holds
( not ( b
1 is_equality & ( for b
2, b
3 being
Variable holds
not b
1 = b
2 '=' b
3 ) ) & ( ex b
2, b
3 being
Variable st b
1 = b
2 '=' b
3 implies b
1 is_equality ) & not ( b
1 is_membership & ( for b
2, b
3 being
Variable holds
not b
1 = b
2 'in' b
3 ) ) & ( ex b
2, b
3 being
Variable st b
1 = b
2 'in' b
3 implies b
1 is_membership ) & not ( b
1 is
negative & ( for b
2 being
ZF-formula holds
not b
1 = 'not' b
2 ) ) & ( ex b
2 being
ZF-formula st b
1 = 'not' b
2 implies b
1 is
negative ) & not ( b
1 is
conjunctive & ( for b
2, b
3 being
ZF-formula holds
not b
1 = b
2 '&' b
3 ) ) & ( ex b
2, b
3 being
ZF-formula st b
1 = b
2 '&' b
3 implies b
1 is
conjunctive ) & not ( b
1 is
universal & ( for b
2 being
Variablefor b
3 being
ZF-formula holds
not b
1 = All b
2,b
3 ) ) & ( ex b
2 being
Variableex b
3 being
ZF-formula st b
1 = All b
2,b
3 implies b
1 is
universal ) )
by Def10, Def11, Def12, Def13, Def14;
:: deftheorem Def15 defines atomic ZF_LANG:def 15 :
:: deftheorem Def16 defines 'or' ZF_LANG:def 16 :
:: deftheorem Def17 defines => ZF_LANG:def 17 :
:: deftheorem Def18 defines <=> ZF_LANG:def 18 :
:: deftheorem Def19 defines Ex ZF_LANG:def 19 :
:: deftheorem Def20 defines disjunctive ZF_LANG:def 20 :
:: deftheorem Def21 defines conditional ZF_LANG:def 21 :
:: deftheorem Def22 defines biconditional ZF_LANG:def 22 :
:: deftheorem Def23 defines existential ZF_LANG:def 23 :
theorem Th17: :: ZF_LANG:17
canceled;
theorem Th18: :: ZF_LANG:18
canceled;
theorem Th19: :: ZF_LANG:19
canceled;
theorem Th20: :: ZF_LANG:20
canceled;
theorem Th21: :: ZF_LANG:21
canceled;
theorem Th22: :: ZF_LANG:22
for b
1 being
ZF-formula holds
( not ( b
1 is
disjunctive & ( for b
2, b
3 being
ZF-formula holds
not b
1 = b
2 'or' b
3 ) ) & ( ex b
2, b
3 being
ZF-formula st b
1 = b
2 'or' b
3 implies b
1 is
disjunctive ) & not ( b
1 is
conditional & ( for b
2, b
3 being
ZF-formula holds
not b
1 = b
2 => b
3 ) ) & ( ex b
2, b
3 being
ZF-formula st b
1 = b
2 => b
3 implies b
1 is
conditional ) & not ( b
1 is
biconditional & ( for b
2, b
3 being
ZF-formula holds
not b
1 = b
2 <=> b
3 ) ) & ( ex b
2, b
3 being
ZF-formula st b
1 = b
2 <=> b
3 implies b
1 is
biconditional ) & not ( b
1 is
existential & ( for b
2 being
Variablefor b
3 being
ZF-formula holds
not b
1 = Ex b
2,b
3 ) ) & ( ex b
2 being
Variableex b
3 being
ZF-formula st b
1 = Ex b
2,b
3 implies b
1 is
existential ) )
by Def20, Def21, Def22, Def23;
definition
let c
1, c
2 be
Variable;
let c
3 be
ZF-formula;
func All c
1,c
2,c
3 -> ZF-formula equals :: ZF_LANG:def 24
All a
1,
(All a2,a3);
coherence
All c1,(All c2,c3) is ZF-formula
;
func Ex c
1,c
2,c
3 -> ZF-formula equals :: ZF_LANG:def 25
Ex a
1,
(Ex a2,a3);
coherence
Ex c1,(Ex c2,c3) is ZF-formula
;
end;
:: deftheorem Def24 defines All ZF_LANG:def 24 :
:: deftheorem Def25 defines Ex ZF_LANG:def 25 :
theorem Th23: :: ZF_LANG:23
definition
let c
1, c
2, c
3 be
Variable;
let c
4 be
ZF-formula;
func All c
1,c
2,c
3,c
4 -> ZF-formula equals :: ZF_LANG:def 26
All a
1,
(All a2,a3,a4);
coherence
All c1,(All c2,c3,c4) is ZF-formula
;
func Ex c
1,c
2,c
3,c
4 -> ZF-formula equals :: ZF_LANG:def 27
Ex a
1,
(Ex a2,a3,a4);
coherence
Ex c1,(Ex c2,c3,c4) is ZF-formula
;
end;
:: deftheorem Def26 defines All ZF_LANG:def 26 :
:: deftheorem Def27 defines Ex ZF_LANG:def 27 :
theorem Th24: :: ZF_LANG:24
for b
1, b
2, b
3 being
Variablefor b
4 being
ZF-formula holds
(
All b
1,b
2,b
3,b
4 = All b
1,
(All b2,b3,b4) &
Ex b
1,b
2,b
3,b
4 = Ex b
1,
(Ex b2,b3,b4) ) ;
theorem Th25: :: ZF_LANG:25
theorem Th26: :: ZF_LANG:26
theorem Th27: :: ZF_LANG:27
theorem Th28: :: ZF_LANG:28
theorem Th29: :: ZF_LANG:29
theorem Th30: :: ZF_LANG:30
theorem Th31: :: ZF_LANG:31
theorem Th32: :: ZF_LANG:32
theorem Th33: :: ZF_LANG:33
theorem Th34: :: ZF_LANG:34
theorem Th35: :: ZF_LANG:35
theorem Th36: :: ZF_LANG:36
theorem Th37: :: ZF_LANG:37
theorem Th38: :: ZF_LANG:38
theorem Th39: :: ZF_LANG:39
theorem Th40: :: ZF_LANG:40
theorem Th41: :: ZF_LANG:41
theorem Th42: :: ZF_LANG:42
theorem Th43: :: ZF_LANG:43
theorem Th44: :: ZF_LANG:44
theorem Th45: :: ZF_LANG:45
theorem Th46: :: ZF_LANG:46
theorem Th47: :: ZF_LANG:47
for b
1, b
2, b
3, b
4 being
ZF-formula holds
( b
1 '&' b
2 = b
3 '&' b
4 implies ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th48: :: ZF_LANG:48
theorem Th49: :: ZF_LANG:49
for b
1, b
2, b
3, b
4 being
ZF-formula holds
( b
1 => b
2 = b
3 => b
4 implies ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th50: :: ZF_LANG:50
for b
1, b
2, b
3, b
4 being
ZF-formula holds
( b
1 <=> b
2 = b
3 <=> b
4 implies ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th51: :: ZF_LANG:51
:: deftheorem Def28 defines Var1 ZF_LANG:def 28 :
:: deftheorem Def29 defines Var2 ZF_LANG:def 29 :
theorem Th52: :: ZF_LANG:52
theorem Th53: :: ZF_LANG:53
theorem Th54: :: ZF_LANG:54
:: deftheorem Def30 defines the_argument_of ZF_LANG:def 30 :
definition
let c
1 be
ZF-formula;
assume E41:
( c
1 is
conjunctive or c
1 is
disjunctive )
;
func the_left_argument_of c
1 -> ZF-formula means :
Def31:
:: ZF_LANG:def 31
ex b
1 being
ZF-formula st a
2 '&' b
1 = a
1 if a
1 is
conjunctive otherwise ex b
1 being
ZF-formula st a
2 'or' b
1 = a
1;
existence
( not ( c1 is conjunctive & ( for b1, b2 being ZF-formula holds
not b1 '&' b2 = c1 ) ) & not ( not c1 is conjunctive & ( for b1, b2 being ZF-formula holds
not b1 'or' b2 = c1 ) ) )
by E41, Def13, Def20;
uniqueness
for b1, b2 being ZF-formula holds
( ( c1 is conjunctive & ex b3 being ZF-formula st b1 '&' b3 = c1 & ex b3 being ZF-formula st b2 '&' b3 = c1 implies b1 = b2 ) & ( not c1 is conjunctive & ex b3 being ZF-formula st b1 'or' b3 = c1 & ex b3 being ZF-formula st b2 'or' b3 = c1 implies b1 = b2 ) )
by Th47, Th48;
consistency
for b1 being ZF-formula holds
verum
;
func the_right_argument_of c
1 -> ZF-formula means :
Def32:
:: ZF_LANG:def 32
ex b
1 being
ZF-formula st b
1 '&' a
2 = a
1 if a
1 is
conjunctive otherwise ex b
1 being
ZF-formula st b
1 'or' a
2 = a
1;
existence
( not ( c1 is conjunctive & ( for b1, b2 being ZF-formula holds
not b2 '&' b1 = c1 ) ) & not ( not c1 is conjunctive & ( for b1, b2 being ZF-formula holds
not b2 'or' b1 = c1 ) ) )
uniqueness
for b1, b2 being ZF-formula holds
( ( c1 is conjunctive & ex b3 being ZF-formula st b3 '&' b1 = c1 & ex b3 being ZF-formula st b3 '&' b2 = c1 implies b1 = b2 ) & ( not c1 is conjunctive & ex b3 being ZF-formula st b3 'or' b1 = c1 & ex b3 being ZF-formula st b3 'or' b2 = c1 implies b1 = b2 ) )
by Th47, Th48;
consistency
for b1 being ZF-formula holds
verum
;
end;
:: deftheorem Def31 defines the_left_argument_of ZF_LANG:def 31 :
:: deftheorem Def32 defines the_right_argument_of ZF_LANG:def 32 :
theorem Th55: :: ZF_LANG:55
canceled;
theorem Th56: :: ZF_LANG:56
theorem Th57: :: ZF_LANG:57
theorem Th58: :: ZF_LANG:58
theorem Th59: :: ZF_LANG:59
definition
let c
1 be
ZF-formula;
assume E45:
( c
1 is
universal or c
1 is
existential )
;
func bound_in c
1 -> Variable means :
Def33:
:: ZF_LANG:def 33
ex b
1 being
ZF-formula st
All a
2,b
1 = a
1 if a
1 is
universal otherwise ex b
1 being
ZF-formula st
Ex a
2,b
1 = a
1;
existence
( not ( c1 is universal & ( for b1 being Variable
for b2 being ZF-formula holds
not All b1,b2 = c1 ) ) & not ( not c1 is universal & ( for b1 being Variable
for b2 being ZF-formula holds
not Ex b1,b2 = c1 ) ) )
by E45, Def14, Def23;
uniqueness
for b1, b2 being Variable holds
( ( c1 is universal & ex b3 being ZF-formula st All b1,b3 = c1 & ex b3 being ZF-formula st All b2,b3 = c1 implies b1 = b2 ) & ( not c1 is universal & ex b3 being ZF-formula st Ex b1,b3 = c1 & ex b3 being ZF-formula st Ex b2,b3 = c1 implies b1 = b2 ) )
by Th12, Th51;
consistency
for b1 being Variable holds
verum
;
func the_scope_of c
1 -> ZF-formula means :
Def34:
:: ZF_LANG:def 34
ex b
1 being
Variable st
All b
1,a
2 = a
1 if a
1 is
universal otherwise ex b
1 being
Variable st
Ex b
1,a
2 = a
1;
existence
( not ( c1 is universal & ( for b1 being ZF-formula
for b2 being Variable holds
not All b2,b1 = c1 ) ) & not ( not c1 is universal & ( for b1 being ZF-formula
for b2 being Variable holds
not Ex b2,b1 = c1 ) ) )
uniqueness
for b1, b2 being ZF-formula holds
( ( c1 is universal & ex b3 being Variable st All b3,b1 = c1 & ex b3 being Variable st All b3,b2 = c1 implies b1 = b2 ) & ( not c1 is universal & ex b3 being Variable st Ex b3,b1 = c1 & ex b3 being Variable st Ex b3,b2 = c1 implies b1 = b2 ) )
by Th12, Th51;
consistency
for b1 being ZF-formula holds
verum
;
end;
:: deftheorem Def33 defines bound_in ZF_LANG:def 33 :
:: deftheorem Def34 defines the_scope_of ZF_LANG:def 34 :
theorem Th60: :: ZF_LANG:60
theorem Th61: :: ZF_LANG:61
theorem Th62: :: ZF_LANG:62
theorem Th63: :: ZF_LANG:63
:: deftheorem Def35 defines the_antecedent_of ZF_LANG:def 35 :
:: deftheorem Def36 defines the_consequent_of ZF_LANG:def 36 :
theorem Th64: :: ZF_LANG:64
theorem Th65: :: ZF_LANG:65
:: deftheorem Def37 defines the_left_side_of ZF_LANG:def 37 :
:: deftheorem Def38 defines the_right_side_of ZF_LANG:def 38 :
theorem Th66: :: ZF_LANG:66
theorem Th67: :: ZF_LANG:67
:: deftheorem Def39 defines is_immediate_constituent_of ZF_LANG:def 39 :
theorem Th68: :: ZF_LANG:68
canceled;
theorem Th69: :: ZF_LANG:69
theorem Th70: :: ZF_LANG:70
theorem Th71: :: ZF_LANG:71
theorem Th72: :: ZF_LANG:72
theorem Th73: :: ZF_LANG:73
theorem Th74: :: ZF_LANG:74
theorem Th75: :: ZF_LANG:75
theorem Th76: :: ZF_LANG:76
theorem Th77: :: ZF_LANG:77
:: deftheorem Def40 defines is_subformula_of ZF_LANG:def 40 :
theorem Th78: :: ZF_LANG:78
canceled;
theorem Th79: :: ZF_LANG:79
:: deftheorem Def41 defines is_proper_subformula_of ZF_LANG:def 41 :
theorem Th80: :: ZF_LANG:80
canceled;
theorem Th81: :: ZF_LANG:81
theorem Th82: :: ZF_LANG:82
theorem Th83: :: ZF_LANG:83
theorem Th84: :: ZF_LANG:84
theorem Th85: :: ZF_LANG:85
theorem Th86: :: ZF_LANG:86
theorem Th87: :: ZF_LANG:87
theorem Th88: :: ZF_LANG:88
theorem Th89: :: ZF_LANG:89
theorem Th90: :: ZF_LANG:90
theorem Th91: :: ZF_LANG:91
theorem Th92: :: ZF_LANG:92
theorem Th93: :: ZF_LANG:93
theorem Th94: :: ZF_LANG:94
theorem Th95: :: ZF_LANG:95
theorem Th96: :: ZF_LANG:96
theorem Th97: :: ZF_LANG:97
theorem Th98: :: ZF_LANG:98
:: deftheorem Def42 defines Subformulae ZF_LANG:def 42 :
theorem Th99: :: ZF_LANG:99
canceled;
theorem Th100: :: ZF_LANG:100
theorem Th101: :: ZF_LANG:101
theorem Th102: :: ZF_LANG:102
theorem Th103: :: ZF_LANG:103
theorem Th104: :: ZF_LANG:104
theorem Th105: :: ZF_LANG:105
theorem Th106: :: ZF_LANG:106
theorem Th107: :: ZF_LANG:107
theorem Th108: :: ZF_LANG:108
theorem Th109: :: ZF_LANG:109
theorem Th110: :: ZF_LANG:110
theorem Th111: :: ZF_LANG:111