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Theorem unex 4459
Description: The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
Hypotheses
Ref Expression
unex.1  |-  A  e. 
_V
unex.2  |-  B  e. 
_V
Assertion
Ref Expression
unex  |-  ( A  u.  B )  e. 
_V

Proof of Theorem unex
StepHypRef Expression
1 unex.1 . . 3  |-  A  e. 
_V
2 unex.2 . . 3  |-  B  e. 
_V
31, 2unipr 3838 . 2  |-  U. { A ,  B }  =  ( A  u.  B )
4 prexg 4229 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
51, 2, 4mp2an 426 . . 3  |-  { A ,  B }  e.  _V
65uniex 4455 . 2  |-  U. { A ,  B }  e.  _V
73, 6eqeltrri 2263 1  |-  ( A  u.  B )  e. 
_V
Colors of variables: wff set class
Syntax hints:    e. wcel 2160   _Vcvv 2752    u. cun 3142   {cpr 3608   U.cuni 3824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-uni 3825
This theorem is referenced by:  unexb  4460  rdg0  6412  unen  6842  findcard2  6917  findcard2s  6918  ac6sfi  6926  sbthlemi10  6995  finomni  7168  exmidfodomrlemim  7230  nn0ex  9212  xrex  9886  exmidunben  12477  strleun  12616  fnpsr  13945
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