Theorem uneqin 3332

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 3156	. . . 4 |- ( ( A u. B ) = ( A i^i B ) -> ( A u. B ) C_ ( A i^i B ) )
2		unss 3255	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) <-> ( A u. B ) C_ ( A i^i B ) )
3		ssin 3303	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) <-> A C_ ( A i^i B ) )
4		sstr 3110	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) -> A C_ B )
5	3, 4	sylbir 134	. . . . . 6 |- ( A C_ ( A i^i B ) -> A C_ B )
6		ssin 3303	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) <-> B C_ ( A i^i B ) )
7		simpl 108	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) -> B C_ A )
8	6, 7	sylbir 134	. . . . . 6 |- ( B C_ ( A i^i B ) -> B C_ A )
9	5, 8	anim12i 336	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) -> ( A C_ B /\ B C_ A ) )
10	2, 9	sylbir 134	. . . 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 3117	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
13	11, 12	sylibr 133	. 2 |- ( ( A u. B ) = ( A i^i B ) -> A = B )
14		unidm 3224	. . . 4 |- ( A u. A ) = A
15		inidm 3290	. . . 4 |- ( A i^i A ) = A
16	14, 15	eqtr4i 2164	. . 3 |- ( A u. A ) = ( A i^i A )
17		uneq2 3229	. . 3 |- ( A = B -> ( A u. A ) = ( A u. B ) )
18		ineq2 3276	. . 3 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
19	16, 17, 18	3eqtr3a 2197	. 2 |- ( A = B -> ( A u. B ) = ( A i^i B ) )
20	13, 19	impbii 125	1 |- ( ( A u. B ) = ( A i^i B ) <-> A = B )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   u. cun 3074   i^i cin 3075   C_ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089

This theorem is referenced by: (None)