Theorem ssunieq 3872

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 3867	. . 3 |- ( A e. B -> A C_ U. B )
2		unissb 3869	. . . 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 3198	. 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 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   U.cuni 3839

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840

This theorem is referenced by:  unimax  3873  hashinfuni  10869  hashennnuni  10871