Theorem ssunieq 3211

Description: Relationship implying union.

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem ssunieq

Step	Hyp	Ref	Expression
1		elssuni 3206	. . 3 |- (A e. B -> A C_ U.B)
2		unissb 3208	. . . 4 |- (U.B C_ A <-> A.x e. B x C_ A)
3	2	biimpri 169	. . 3 |- (A.x e. B x C_ A -> U.B C_ A)
4	1, 3	anim12i 360	. 2 |- ((A e. B /\ A.x e. B x C_ A) -> (A C_ U.B /\ U.B C_ A))
5		eqss 2631	. 2 |- (A = U.B <-> (A C_ U.B /\ U.B C_ A))
6	4, 5	sylibr 217	1 |- ((A e. B /\ A.x e. B x C_ A) -> A = U.B)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  U.cuni 3177

This theorem is referenced by:  unimax 3212  shsspwh 10751

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-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178

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