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Theorem unexg 4204
Description: A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
Assertion
Ref Expression
unexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  e.  _V )

Proof of Theorem unexg
StepHypRef Expression
1 elex 2611 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2611 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 unexb 4203 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  <->  ( A  u.  B )  e.  _V )
43biimpi 118 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  u.  B
)  e.  _V )
51, 2, 4syl2an 283 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1434   _Vcvv 2602    u. cun 2972
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 2064  ax-sep 3904  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-uni 3610
This theorem is referenced by:  tpexg  4205  eldifpw  4234  xpexg  4480  tposexg  5907  tfrlemisucaccv  5974  tfrlemibxssdm  5976  tfrlemibfn  5977  tfr1onlemsucaccv  5990  tfr1onlembxssdm  5992  tfr1onlembfn  5993  tfrcllemsucaccv  6003  tfrcllembxssdm  6005  tfrcllembfn  6006  rdgtfr  6023  rdgruledefgg  6024  rdgivallem  6030
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