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Theorem uniexb 4233
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 4203 . 2  |-  ( A  e.  _V  ->  U. A  e.  _V )
2 pwuni 3971 . . 3  |-  A  C_  ~P U. A
3 pwexg 3961 . . 3  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
4 ssexg 3924 . . 3  |-  ( ( A  C_  ~P U. A  /\  ~P U. A  e. 
_V )  ->  A  e.  _V )
52, 3, 4sylancr 399 . 2  |-  ( U. A  e.  _V  ->  A  e.  _V )
61, 5impbii 121 1  |-  ( A  e.  _V  <->  U. A  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 102    e. wcel 1409   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387   U.cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-uni 3609
This theorem is referenced by:  pwexb  4234  tfrlemibex  5974
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