Theorem unissel 4468

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 473	. 2 |- ( ( U. A C_ B /\ B e. A ) -> U. A C_ B )
2		elssuni 4467	. . 3 |- ( B e. A -> B C_ U. A )
3	2	adantl 482	. 2 |- ( ( U. A C_ B /\ B e. A ) -> B C_ U. A )
4	1, 3	eqssd 3620	1 |- ( ( U. A C_ B /\ B e. A ) -> U. A = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   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-v 3202  df-in 3581  df-ss 3588  df-uni 4437

This theorem is referenced by:  elpwuni  4616  mretopd  20896  toponmre  20897  neiptopuni  20934  filunibas  21685  unidmvol  23309  unicls  29949  carsguni  30370