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Theorem uniexb 4323
Description: The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexb  |-  ( A  e.  _V  <->  U. A  e. 
_V )

Proof of Theorem uniexb
StepHypRef Expression
1 uniexg 4290 . 2  |-  ( A  e.  _V  ->  U. A  e.  _V )
2 pwuni 4048 . . 3  |-  A  C_  ~P U. A
3 pwexg 4036 . . 3  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
4 ssexg 3999 . . 3  |-  ( ( A  C_  ~P U. A  /\  ~P U. A  e. 
_V )  ->  A  e.  _V )
52, 3, 4sylancr 406 . 2  |-  ( U. A  e.  _V  ->  A  e.  _V )
61, 5impbii 125 1  |-  ( A  e.  _V  <->  U. A  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1445   _Vcvv 2633    C_ wss 3013   ~Pcpw 3449   U.cuni 3675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-pw 3451  df-uni 3676
This theorem is referenced by:  pwexb  4324  elpwpwel  4325  tfrlemibex  6132  tfr1onlembex  6148  tfrcllembex  6161  ixpexgg  6519  tgss2  11947
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