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Theorem unex 4222
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 3635 . 2  |-  U. { A ,  B }  =  ( A  u.  B )
4 prexg 3994 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
51, 2, 4mp2an 417 . . 3  |-  { A ,  B }  e.  _V
65uniex 4220 . 2  |-  U. { A ,  B }  e.  _V
73, 6eqeltrri 2156 1  |-  ( A  u.  B )  e. 
_V
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   _Vcvv 2610    u. cun 2980   {cpr 3417   U.cuni 3621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-uni 3622
This theorem is referenced by:  unexb  4223  rdg0  6056  unen  6382  findcard2  6445  findcard2s  6446  ac6sfi  6454  nn0ex  8413  xrex  9038
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