Theorem unissel 2517

Description: Condition turning a subclass relationship for union into an equality.

Assertion

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

Proof of Theorem unissel

Step	Hyp	Ref	Expression
1		pm3.26 319	. 2 |- ((U.A (_ B /\ B e. A) -> U.A (_ B)
2		elssuni 2516	. . 3 |- (B e. A -> B (_ U.A)
3	2	adantl 388	. 2 |- ((U.A (_ B /\ B e. A) -> B (_ U.A)
4	1, 3	eqssd 2069	1 |- ((U.A (_ B /\ B e. A) -> U.A = B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   (_ wss 2037  U.cuni 2493

This theorem is referenced by:  elpwuni 2751  istps2 7549  unnei 7676

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041  df-ss 2043  df-uni 2494

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