Theorem uniexb 2907

Description: The Axiom of Union and its converse. A class is a set iff its union is a set.

Assertion

Ref	Expression
uniexb	|- (A e. V <-> U.A e. V)

Proof of Theorem uniexb

Step	Hyp	Ref	Expression
1		uniexg 2871	. 2 |- (A e. V -> U.A e. V)
2		pwuni 2757	. . 3 |- A (_ P~U.A
3		ssexg 2721	. . . 4 |- ((A (_ P~U.A /\ P~U.A e. V) -> A e. V)
4		pwexg 2746	. . . 4 |- (U.A e. V -> P~U.A e. V)
5	3, 4	sylan2 451	. . 3 |- ((A (_ P~U.A /\ U.A e. V) -> A e. V)
6	2, 5	mpan 695	. 2 |- (U.A e. V -> A e. V)
7	1, 6	impbi 157	1 |- (A e. V <-> U.A e. V)

Syntax hints:   <-> wb 146   e. wcel 958  Vcvv 1811   (_ wss 2047  P~cpw 2401  U.cuni 2503

This theorem is referenced by:  pwexb 2908  ixpexg 4358  pwuninel 4486  rankuni 4698  unialeph 4895  eltopsp 7604

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-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-uni 2504

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