Theorem ssunieq 3634

Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Assertion

Ref	Expression
ssunieq	|- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem ssunieq

Step	Hyp	Ref	Expression
1		elssuni 3629	. . 3 |- ( A e. B -> A C_ U. B )
2		unissb 3631	. . . 4 |- ( U. B C_ A <-> A. x e. B x C_ A )
3	2	biimpri 131	. . 3 |- ( A. x e. B x C_ A -> U. B C_ A )
4	1, 3	anim12i 331	. 2 |- ( ( A e. B /\ A. x e. B x C_ A ) -> ( A C_ U. B /\ U. B C_ A ) )
5		eqss 3014	. 2 |- ( A = U. B <-> ( A C_ U. B /\ U. B C_ A ) )
6	4, 5	sylibr 132	1 |- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   C_ wss 2973   U.cuni 3601

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-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602

This theorem is referenced by:  unimax  3635