Theorem uneqin 3188

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 2997	. . . 4 |- ( ( A u. B ) = ( A i^i B ) -> ( A u. B ) C_ ( A i^i B ) )
2		unss 3117	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) <-> ( A u. B ) C_ ( A i^i B ) )
3		ssin 3159	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) <-> A C_ ( A i^i B ) )
4		sstr 2953	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) -> A C_ B )
5	3, 4	sylbir 125	. . . . . 6 |- ( A C_ ( A i^i B ) -> A C_ B )
6		ssin 3159	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) <-> B C_ ( A i^i B ) )
7		simpl 102	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) -> B C_ A )
8	6, 7	sylbir 125	. . . . . 6 |- ( B C_ ( A i^i B ) -> B C_ A )
9	5, 8	anim12i 321	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) -> ( A C_ B /\ B C_ A ) )
10	2, 9	sylbir 125	. . . 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 2960	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
13	11, 12	sylibr 137	. 2 |- ( ( A u. B ) = ( A i^i B ) -> A = B )
14		unidm 3086	. . . 4 |- ( A u. A ) = A
15		inidm 3146	. . . 4 |- ( A i^i A ) = A
16	14, 15	eqtr4i 2063	. . 3 |- ( A u. A ) = ( A i^i A )
17		uneq2 3091	. . 3 |- ( A = B -> ( A u. A ) = ( A u. B ) )
18		ineq2 3132	. . 3 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
19	16, 17, 18	3eqtr3a 2096	. 2 |- ( A = B -> ( A u. B ) = ( A i^i B ) )
20	13, 19	impbii 117	1 |- ( ( A u. B ) = ( A i^i B ) <-> A = B )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   u. cun 2915   i^i cin 2916   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931

This theorem is referenced by: (None)