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Theorem unex 4424
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 3808 . 2  |-  U. { A ,  B }  =  ( A  u.  B )
4 prexg 4194 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
51, 2, 4mp2an 424 . . 3  |-  { A ,  B }  e.  _V
65uniex 4420 . 2  |-  U. { A ,  B }  e.  _V
73, 6eqeltrri 2244 1  |-  ( A  u.  B )  e. 
_V
Colors of variables: wff set class
Syntax hints:    e. wcel 2141   _Vcvv 2730    u. cun 3119   {cpr 3582   U.cuni 3794
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-uni 3795
This theorem is referenced by:  unexb  4425  rdg0  6364  unen  6791  findcard2  6864  findcard2s  6865  ac6sfi  6873  sbthlemi10  6940  finomni  7113  exmidfodomrlemim  7167  nn0ex  9130  xrex  9802  exmidunben  12370  strleun  12496
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