Theorem unissel 3801

Description: Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)

Assertion

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

Proof of Theorem unissel

Step	Hyp	Ref	Expression
1		simpl 108	. 2 |- ( ( U. A C_ B /\ B e. A ) -> U. A C_ B )
2		elssuni 3800	. . 3 |- ( B e. A -> B C_ U. A )
3	2	adantl 275	. 2 |- ( ( U. A C_ B /\ B e. A ) -> B C_ U. A )
4	1, 3	eqssd 3145	1 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   C_ wss 3102   U.cuni 3772

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773

This theorem is referenced by:  elpwuni  3938