Theorem ssunieq 4472

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 4467	. . 3 |- ( A e. B -> A C_ U. B )
2		unissb 4469	. . . 4 |- ( U. B C_ A <-> A. x e. B x C_ A )
3	2	biimpri 218	. . 3 |- ( A. x e. B x C_ A -> U. B C_ A )
4	1, 3	anim12i 590	. 2 |- ( ( A e. B /\ A. x e. B x C_ A ) -> ( A C_ U. B /\ U. B C_ A ) )
5		eqss 3618	. 2 |- ( A = U. B <-> ( A C_ U. B /\ U. B C_ A ) )
6	4, 5	sylibr 224	1 |- ( ( A e. B /\ A. x e. B x C_ A ) -> A = U. B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   U.cuni 4436

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-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437

This theorem is referenced by:  unimax  4473  shsspwh  28103