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Theorem bj-unexg 10041
Description: unexg 4178 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-unexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  e.  _V )

Proof of Theorem bj-unexg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3090 . . 3  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
2 eleq1 2100 . . 3  |-  ( ( x  u.  y )  =  ( A  u.  y )  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
31, 2syl 14 . 2  |-  ( x  =  A  ->  (
( x  u.  y
)  e.  _V  <->  ( A  u.  y )  e.  _V ) )
4 uneq2 3091 . . 3  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
5 eleq1 2100 . . 3  |-  ( ( A  u.  y )  =  ( A  u.  B )  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
64, 5syl 14 . 2  |-  ( y  =  B  ->  (
( A  u.  y
)  e.  _V  <->  ( A  u.  B )  e.  _V ) )
7 vex 2560 . . 3  |-  x  e. 
_V
8 vex 2560 . . 3  |-  y  e. 
_V
97, 8bj-unex 10039 . 2  |-  ( x  u.  y )  e. 
_V
103, 6, 9vtocl2g 2617 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   _Vcvv 2557    u. cun 2915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-bdc 9961
This theorem is referenced by:  bj-sucexg  10042
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