:: SUBSET_1 semantic presentation
registration
let x1 be
set ;
cluster {a1} -> non
empty ;
coherence
not {x1} is empty
by TARSKI:def 1;
let x2 be
set ;
cluster {a1,a2} -> non
empty ;
coherence
not {x1,x2} is empty
by TARSKI:def 2;
let x3 be
set ;
cluster {a1,a2,a3} -> non
empty ;
coherence
not {x1,x2,x3} is empty
by ENUMSET1:def 1;
let x4 be
set ;
cluster {a1,a2,a3,a4} -> non
empty ;
coherence
not {x1,x2,x3,x4} is empty
by ENUMSET1:def 2;
let x5 be
set ;
cluster {a1,a2,a3,a4,a5} -> non
empty ;
coherence
not {x1,x2,x3,x4,x5} is empty
by ENUMSET1:def 3;
let x6 be
set ;
cluster {a1,a2,a3,a4,a5,a6} -> non
empty ;
coherence
not {x1,x2,x3,x4,x5,x6} is empty
by ENUMSET1:def 4;
let x7 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7} -> non
empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7} is empty
by ENUMSET1:def 5;
let x8 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8} -> non
empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7,x8} is empty
by ENUMSET1:def 6;
let x9 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8,a9} -> non
empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7,x8,x9} is empty
by ENUMSET1:def 7;
let x10 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8,a9,a10} -> non
empty ;
coherence
not {x1,x2,x3,x4,x5,x6,x7,x8,x9,x10} is empty
by ENUMSET1:def 8;
end;
:: deftheorem Def1 SUBSET_1:def 1 :
canceled;
:: deftheorem Def2 defines Element SUBSET_1:def 2 :
Lemma27:
for E being set
for X being Subset of E
for x being set st x in X holds
x in E
:: deftheorem Def3 defines {} SUBSET_1:def 3 :
:: deftheorem Def4 defines [#] SUBSET_1:def 4 :
theorem Th1: :: SUBSET_1:1
canceled;
theorem Th2: :: SUBSET_1:2
canceled;
theorem Th3: :: SUBSET_1:3
canceled;
theorem Th4: :: SUBSET_1:4
theorem Th5: :: SUBSET_1:5
canceled;
theorem Th6: :: SUBSET_1:6
canceled;
theorem Th7: :: SUBSET_1:7
for
E being
set for
A,
B being
Subset of
E st ( for
x being
Element of
E st
x in A holds
x in B ) holds
A c= B
theorem Th8: :: SUBSET_1:8
for
E being
set for
A,
B being
Subset of
E st ( for
x being
Element of
E holds
(
x in A iff
x in B ) ) holds
A = B
theorem Th9: :: SUBSET_1:9
canceled;
theorem Th10: :: SUBSET_1:10
:: deftheorem Def5 defines ` SUBSET_1:def 5 :
theorem Th11: :: SUBSET_1:11
canceled;
theorem Th12: :: SUBSET_1:12
canceled;
theorem Th13: :: SUBSET_1:13
canceled;
theorem Th14: :: SUBSET_1:14
canceled;
theorem Th15: :: SUBSET_1:15
for
E being
set for
A,
B,
C being
Subset of
E st ( for
x being
Element of
E holds
(
x in A iff (
x in B or
x in C ) ) ) holds
A = B \/ C
theorem Th16: :: SUBSET_1:16
for
E being
set for
A,
B,
C being
Subset of
E st ( for
x being
Element of
E holds
(
x in A iff (
x in B &
x in C ) ) ) holds
A = B /\ C
theorem Th17: :: SUBSET_1:17
for
E being
set for
A,
B,
C being
Subset of
E st ( for
x being
Element of
E holds
(
x in A iff (
x in B & not
x in C ) ) ) holds
A = B \ C
theorem Th18: :: SUBSET_1:18
for
E being
set for
A,
B,
C being
Subset of
E st ( for
x being
Element of
E holds
(
x in A iff ( (
x in B & not
x in C ) or (
x in C & not
x in B ) ) ) ) holds
A = B \+\ C
theorem Th19: :: SUBSET_1:19
canceled;
theorem Th20: :: SUBSET_1:20
canceled;
theorem Th21: :: SUBSET_1:21
theorem Th22: :: SUBSET_1:22
theorem Th23: :: SUBSET_1:23
canceled;
theorem Th24: :: SUBSET_1:24
canceled;
theorem Th25: :: SUBSET_1:25
theorem Th26: :: SUBSET_1:26
theorem Th27: :: SUBSET_1:27
canceled;
theorem Th28: :: SUBSET_1:28
theorem Th29: :: SUBSET_1:29
theorem Th30: :: SUBSET_1:30
theorem Th31: :: SUBSET_1:31
theorem Th32: :: SUBSET_1:32
theorem Th33: :: SUBSET_1:33
theorem Th34: :: SUBSET_1:34
theorem Th35: :: SUBSET_1:35
theorem Th36: :: SUBSET_1:36
theorem Th37: :: SUBSET_1:37
canceled;
theorem Th38: :: SUBSET_1:38
theorem Th39: :: SUBSET_1:39
theorem Th40: :: SUBSET_1:40
theorem Th41: :: SUBSET_1:41
theorem Th42: :: SUBSET_1:42
theorem Th43: :: SUBSET_1:43
theorem Th44: :: SUBSET_1:44
theorem Th45: :: SUBSET_1:45
canceled;
theorem Th46: :: SUBSET_1:46
theorem Th47: :: SUBSET_1:47
theorem Th48: :: SUBSET_1:48
for
E being
set for
A,
B being
Subset of
E st ( for
a being
Element of
A holds
a in B ) holds
A c= B
theorem Th49: :: SUBSET_1:49
for
E being
set for
A being
Subset of
E st ( for
x being
Element of
E holds
x in A ) holds
E = A
theorem Th50: :: SUBSET_1:50
theorem Th51: :: SUBSET_1:51
for
E being
set for
A,
B being
Subset of
E st ( for
x being
Element of
E holds
(
x in A iff not
x in B ) ) holds
A = B `
theorem Th52: :: SUBSET_1:52
for
E being
set for
A,
B being
Subset of
E st ( for
x being
Element of
E holds
( not
x in A iff
x in B ) ) holds
A = B `
theorem Th53: :: SUBSET_1:53
for
E being
set for
A,
B being
Subset of
E st ( for
x being
Element of
E holds
( (
x in A & not
x in B ) or (
x in B & not
x in A ) ) ) holds
A = B `
theorem Th54: :: SUBSET_1:54
theorem Th55: :: SUBSET_1:55
theorem Th56: :: SUBSET_1:56
theorem Th57: :: SUBSET_1:57
theorem Th58: :: SUBSET_1:58
theorem Th59: :: SUBSET_1:59
theorem Th60: :: SUBSET_1:60
for
X being
set for
x1,
x2,
x3,
x4,
x5,
x6 being
Element of
X st
X <> {} holds
{x1,x2,x3,x4,x5,x6} is
Subset of
X
theorem Th61: :: SUBSET_1:61
for
X being
set for
x1,
x2,
x3,
x4,
x5,
x6,
x7 being
Element of
X st
X <> {} holds
{x1,x2,x3,x4,x5,x6,x7} is
Subset of
X
theorem Th62: :: SUBSET_1:62
for
X being
set for
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8 being
Element of
X st
X <> {} holds
{x1,x2,x3,x4,x5,x6,x7,x8} is
Subset of
X
theorem Th63: :: SUBSET_1:63
:: deftheorem Def6 defines choose SUBSET_1:def 6 :
for
S being
set st contradiction holds
for
b2 being
Element of
S holds
(
b2 = choose S iff verum );
Lemma49:
for X, Y being set st ( for x being set st x in X holds
x in Y ) holds
X is Subset of Y
Lemma50:
for x, E being set
for A being Subset of E st x in A holds
x is Element of E
theorem Th64: :: SUBSET_1:64
for
E being
set for
A,
B being
Subset of
E st
A ` = B ` holds
A = B