Theorem uneqin 3215

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 3051	. . . 4 |- ( ( A u. B ) = ( A i^i B ) -> ( A u. B ) C_ ( A i^i B ) )
2		unss 3146	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) <-> ( A u. B ) C_ ( A i^i B ) )
3		ssin 3188	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) <-> A C_ ( A i^i B ) )
4		sstr 3007	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) -> A C_ B )
5	3, 4	sylbir 133	. . . . . 6 |- ( A C_ ( A i^i B ) -> A C_ B )
6		ssin 3188	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) <-> B C_ ( A i^i B ) )
7		simpl 107	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) -> B C_ A )
8	6, 7	sylbir 133	. . . . . 6 |- ( B C_ ( A i^i B ) -> B C_ A )
9	5, 8	anim12i 331	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) -> ( A C_ B /\ B C_ A ) )
10	2, 9	sylbir 133	. . . 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 3014	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
13	11, 12	sylibr 132	. 2 |- ( ( A u. B ) = ( A i^i B ) -> A = B )
14		unidm 3115	. . . 4 |- ( A u. A ) = A
15		inidm 3175	. . . 4 |- ( A i^i A ) = A
16	14, 15	eqtr4i 2104	. . 3 |- ( A u. A ) = ( A i^i A )
17		uneq2 3120	. . 3 |- ( A = B -> ( A u. A ) = ( A u. B ) )
18		ineq2 3161	. . 3 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
19	16, 17, 18	3eqtr3a 2137	. 2 |- ( A = B -> ( A u. B ) = ( A i^i B ) )
20	13, 19	impbii 124	1 |- ( ( A u. B ) = ( A i^i B ) <-> A = B )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   u. cun 2971   i^i cin 2972   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by: (None)