Theorem unissel 3609

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 102	. 2 |- ( ( U. A C_ B /\ B e. A ) -> U. A C_ B )
2		elssuni 3608	. . 3 |- ( B e. A -> B C_ U. A )
3	2	adantl 262	. 2 |- ( ( U. A C_ B /\ B e. A ) -> B C_ U. A )
4	1, 3	eqssd 2962	1 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   C_ wss 2917   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581

This theorem is referenced by:  elpwuni  3741