Theorem ssunieq 3829

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 3824	. . 3 |- ( A e. B -> A C_ U. B )
2		unissb 3826	. . . 4 |- ( U. B C_ A <-> A. x e. B x C_ A )
3	2	biimpri 132	. . 3 |- ( A. x e. B x C_ A -> U. B C_ A )
4	1, 3	anim12i 336	. 2 |- ( ( A e. B /\ A. x e. B x C_ A ) -> ( A C_ U. B /\ U. B C_ A ) )
5		eqss 3162	. 2 |- ( A = U. B <-> ( A C_ U. B /\ U. B C_ A ) )
6	4, 5	sylibr 133	1 |- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   U.cuni 3796

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797

This theorem is referenced by:  unimax  3830  hashinfuni  10711  hashennnuni  10713