Theorem uneqin 3457

Description: Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem uneqin

Step	Hyp	Ref	Expression
1		eqimss 3280	. . . 4 |- ( ( A u. B ) = ( A i^i B ) -> ( A u. B ) C_ ( A i^i B ) )
2		unss 3380	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) <-> ( A u. B ) C_ ( A i^i B ) )
3		ssin 3428	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) <-> A C_ ( A i^i B ) )
4		sstr 3234	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) -> A C_ B )
5	3, 4	sylbir 135	. . . . . 6 |- ( A C_ ( A i^i B ) -> A C_ B )
6		ssin 3428	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) <-> B C_ ( A i^i B ) )
7		simpl 109	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) -> B C_ A )
8	6, 7	sylbir 135	. . . . . 6 |- ( B C_ ( A i^i B ) -> B C_ A )
9	5, 8	anim12i 338	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) -> ( A C_ B /\ B C_ A ) )
10	2, 9	sylbir 135	. . . 4 |- ( ( A u. B ) C_ ( A i^i B ) -> ( A C_ B /\ B C_ A ) )
11	1, 10	syl 14	. . 3 |- ( ( A u. B ) = ( A i^i B ) -> ( A C_ B /\ B C_ A ) )
12		eqss 3241	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
13	11, 12	sylibr 134	. 2 |- ( ( A u. B ) = ( A i^i B ) -> A = B )
14		unidm 3349	. . . 4 |- ( A u. A ) = A
15		inidm 3415	. . . 4 |- ( A i^i A ) = A
16	14, 15	eqtr4i 2254	. . 3 |- ( A u. A ) = ( A i^i A )
17		uneq2 3354	. . 3 |- ( A = B -> ( A u. A ) = ( A u. B ) )
18		ineq2 3401	. . 3 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
19	16, 17, 18	3eqtr3a 2287	. 2 |- ( A = B -> ( A u. B ) = ( A i^i B ) )
20	13, 19	impbii 126	1 |- ( ( A u. B ) = ( A i^i B ) <-> A = B )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   u. cun 3197   i^i cin 3198   C_ wss 3199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-un 3203  df-in 3205  df-ss 3212

This theorem is referenced by: (None)