Theorem unissel 3879

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 3878	. . 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 3210	1 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   C_ wss 3166   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-uni 3851

This theorem is referenced by:  elpwuni  4017