Theorem unissel 3207

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

Assertion

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

Proof of Theorem unissel

Step	Hyp	Ref	Expression
1		simpl 346	. 2 |- ((U.A C_ B /\ B e. A) -> U.A C_ B)
2		elssuni 3206	. . 3 |- (B e. A -> B C_ U.A)
3	2	adantl 424	. 2 |- ((U.A C_ B /\ B e. A) -> B C_ U.A)
4	1, 3	eqssd 2633	1 |- ((U.A C_ B /\ B e. A) -> U.A = B)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  U.cuni 3177

This theorem is referenced by:  elpwuni 3335  istps2 8876  unnei 9011  heiborlem11 15965

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178

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