Theorem uniexb 2897

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 2862	. 2 |- (A e. V -> U.A e. V)
2		pwuni 2747	. . 3 |- A (_ P~U.A
3		ssexg 2711	. . . 4 |- ((A (_ P~U.A /\ P~U.A e. V) -> A e. V)
4		pwexg 2736	. . . 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 693	. 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 955  Vcvv 1802   (_ wss 2037  P~cpw 2391  U.cuni 2493

This theorem is referenced by:  pwexb 2898  ixpexg 4342  rankuni 4670  unialeph 4867  eltopsp 7546

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-13 966  ax-14 967  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  ax-sep 2693  ax-pow 2732  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  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-pw 2392  df-uni 2494

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