Theorem uneqin 2253

Description: Equality of union and intersection implies equality of their arguments.

Assertion

Ref	Expression
uneqin	|- ((A u. B) = (A i^i B) <-> A = B)

Proof of Theorem uneqin

Step	Hyp	Ref	Expression
1		eqimss 2106	. . 3 |- ((A u. B) = (A i^i B) -> (A u. B) (_ (A i^i B))
2		unss 2201	. . . . 5 |- ((A (_ (A i^i B) /\ B (_ (A i^i B)) <-> (A u. B) (_ (A i^i B))
3		ssin 2229	. . . . . . 7 |- ((A (_ A /\ A (_ B) <-> A (_ (A i^i B))
4		sstr 2069	. . . . . . 7 |- ((A (_ A /\ A (_ B) -> A (_ B)
5	3, 4	sylbir 201	. . . . . 6 |- (A (_ (A i^i B) -> A (_ B)
6		ssin 2229	. . . . . . 7 |- ((B (_ A /\ B (_ B) <-> B (_ (A i^i B))
7		pm3.26 319	. . . . . . 7 |- ((B (_ A /\ B (_ B) -> B (_ A)
8	6, 7	sylbir 201	. . . . . 6 |- (B (_ (A i^i B) -> B (_ A)
9	5, 8	anim12i 333	. . . . 5 |- ((A (_ (A i^i B) /\ B (_ (A i^i B)) -> (A (_ B /\ B (_ A))
10	2, 9	sylbir 201	. . . 4 |- ((A u. B) (_ (A i^i B) -> (A (_ B /\ B (_ A))
11		eqss 2074	. . . 4 |- (A = B <-> (A (_ B /\ B (_ A))
12	10, 11	sylibr 200	. . 3 |- ((A u. B) (_ (A i^i B) -> A = B)
13	1, 12	syl 10	. 2 |- ((A u. B) = (A i^i B) -> A = B)
14		uneq2 2175	. . 3 |- (A = B -> (A u. A) = (A u. B))
15		ineq2 2208	. . . . 5 |- (A = B -> (A i^i A) = (A i^i B))
16		inidm 2219	. . . . 5 |- (A i^i A) = A
17	15, 16	syl5eqr 1519	. . . 4 |- (A = B -> A = (A i^i B))
18		unidm 2172	. . . 4 |- (A u. A) = A
19	17, 18	syl5eq 1517	. . 3 |- (A = B -> (A u. A) = (A i^i B))
20	14, 19	eqtr3d 1507	. 2 |- (A = B -> (A u. B) = (A i^i B))
21	13, 20	impbi 157	1 |- ((A u. B) = (A i^i B) <-> A = B)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 955   u. cun 2042   i^i cin 2043   (_ wss 2044

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-un 2047  df-in 2048  df-ss 2050

Copyright terms: Public domain