Theorem ssunieq 3921

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 3916	. . 3 |- ( A e. B -> A C_ U. B )
2		unissb 3918	. . . 4 |- ( U. B C_ A <-> A. x e. B x C_ A )
3	2	biimpri 133	. . 3 |- ( A. x e. B x C_ A -> U. B C_ A )
4	1, 3	anim12i 338	. 2 |- ( ( A e. B /\ A. x e. B x C_ A ) -> ( A C_ U. B /\ U. B C_ A ) )
5		eqss 3239	. 2 |- ( A = U. B <-> ( A C_ U. B /\ U. B C_ A ) )
6	4, 5	sylibr 134	1 |- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   U.cuni 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-uni 3889

This theorem is referenced by:  unimax  3922  hashinfuni  10999  hashennnuni  11001