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Theorem uniexb 4295
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 4265 . 2  |-  ( A  e.  _V  ->  U. A  e.  _V )
2 pwuni 4027 . . 3  |-  A  C_  ~P U. A
3 pwexg 4015 . . 3  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
4 ssexg 3978 . . 3  |-  ( ( A  C_  ~P U. A  /\  ~P U. A  e. 
_V )  ->  A  e.  _V )
52, 3, 4sylancr 405 . 2  |-  ( U. A  e.  _V  ->  A  e.  _V )
61, 5impbii 124 1  |-  ( A  e.  _V  <->  U. A  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   _Vcvv 2619    C_ wss 2999   ~Pcpw 3429   U.cuni 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-pw 3431  df-uni 3654
This theorem is referenced by:  pwexb  4296  elpwpwel  4297  tfrlemibex  6094  tfr1onlembex  6110  tfrcllembex  6123
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