Theorem uneqin 3878

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 3657	. . . 4 |- ( ( A u. B ) = ( A i^i B ) -> ( A u. B ) C_ ( A i^i B ) )
2		unss 3787	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) <-> ( A u. B ) C_ ( A i^i B ) )
3		ssin 3835	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) <-> A C_ ( A i^i B ) )
4		sstr 3611	. . . . . . 7 |- ( ( A C_ A /\ A C_ B ) -> A C_ B )
5	3, 4	sylbir 225	. . . . . 6 |- ( A C_ ( A i^i B ) -> A C_ B )
6		ssin 3835	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) <-> B C_ ( A i^i B ) )
7		simpl 473	. . . . . . 7 |- ( ( B C_ A /\ B C_ B ) -> B C_ A )
8	6, 7	sylbir 225	. . . . . 6 |- ( B C_ ( A i^i B ) -> B C_ A )
9	5, 8	anim12i 590	. . . . 5 |- ( ( A C_ ( A i^i B ) /\ B C_ ( A i^i B ) ) -> ( A C_ B /\ B C_ A ) )
10	2, 9	sylbir 225	. . . 4 |- ( ( A u. B ) C_ ( A i^i B ) -> ( A C_ B /\ B C_ A ) )
11	1, 10	syl 17	. . 3 |- ( ( A u. B ) = ( A i^i B ) -> ( A C_ B /\ B C_ A ) )
12		eqss 3618	. . 3 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
13	11, 12	sylibr 224	. 2 |- ( ( A u. B ) = ( A i^i B ) -> A = B )
14		unidm 3756	. . . 4 |- ( A u. A ) = A
15		inidm 3822	. . . 4 |- ( A i^i A ) = A
16	14, 15	eqtr4i 2647	. . 3 |- ( A u. A ) = ( A i^i A )
17		uneq2 3761	. . 3 |- ( A = B -> ( A u. A ) = ( A u. B ) )
18		ineq2 3808	. . 3 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
19	16, 17, 18	3eqtr3a 2680	. 2 |- ( A = B -> ( A u. B ) = ( A i^i B ) )
20	13, 19	impbii 199	1 |- ( ( A u. B ) = ( A i^i B ) <-> A = B )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   u. cun 3572   i^i cin 3573   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by:  uniintsn  4514