Theorem unissel 3864

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 109	. 2 |- ( ( U. A C_ B /\ B e. A ) -> U. A C_ B )
2		elssuni 3863	. . 3 |- ( B e. A -> B C_ U. A )
3	2	adantl 277	. 2 |- ( ( U. A C_ B /\ B e. A ) -> B C_ U. A )
4	1, 3	eqssd 3196	1 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3153   U.cuni 3835

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836

This theorem is referenced by:  elpwuni  4002