Theorem ssunieq 2531

Description: Relationship implying union.

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem ssunieq

Step	Hyp	Ref	Expression
1		elssuni 2526	. . 3 |- (A e. B -> A (_ U.B)
2		unissb 2528	. . . 4 |- (U.B (_ A <-> A.x e. B x (_ A)
3	2	biimpr 152	. . 3 |- (A.x e. B x (_ A -> U.B (_ A)
4	1, 3	anim12i 333	. 2 |- ((A e. B /\ A.x e. B x (_ A) -> (A (_ U.B /\ U.B (_ A))
5		eqss 2077	. 2 |- (A = U.B <-> (A (_ U.B /\ U.B (_ A))
6	4, 5	sylibr 200	1 |- ((A e. B /\ A.x e. B x (_ A) -> A = U.B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  U.cuni 2503

This theorem is referenced by:  unimax 2532  shsspwh 9118

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504

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