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Theorem uniexb 4433
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 4399 . 2  |-  ( A  e.  _V  ->  U. A  e.  _V )
2 pwuni 4153 . . 3  |-  A  C_  ~P U. A
3 pwexg 4141 . . 3  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
4 ssexg 4103 . . 3  |-  ( ( A  C_  ~P U. A  /\  ~P U. A  e. 
_V )  ->  A  e.  _V )
52, 3, 4sylancr 411 . 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 2128   _Vcvv 2712    C_ wss 3102   ~Pcpw 3543   U.cuni 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-uni 3773
This theorem is referenced by:  pwexb  4434  elpwpwel  4435  tfrlemibex  6276  tfr1onlembex  6292  tfrcllembex  6305  ixpexgg  6667  tgss2  12479  txbasex  12657
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