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Theorem unex 4330
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 3718 . 2  |-  U. { A ,  B }  =  ( A  u.  B )
4 prexg 4101 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
51, 2, 4mp2an 420 . . 3  |-  { A ,  B }  e.  _V
65uniex 4327 . 2  |-  U. { A ,  B }  e.  _V
73, 6eqeltrri 2189 1  |-  ( A  u.  B )  e. 
_V
Colors of variables: wff set class
Syntax hints:    e. wcel 1463   _Vcvv 2658    u. cun 3037   {cpr 3496   U.cuni 3704
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-uni 3705
This theorem is referenced by:  unexb  4331  rdg0  6250  unen  6676  findcard2  6749  findcard2s  6750  ac6sfi  6758  sbthlemi10  6820  finomni  6978  exmidfodomrlemim  7021  nn0ex  8934  xrex  9579  exmidunben  11834  strleun  11943
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