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Theorem bj-uniex 13145
Description: uniex 4359 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-uniex.1  |-  A  e. 
_V
Assertion
Ref Expression
bj-uniex  |-  U. A  e.  _V

Proof of Theorem bj-uniex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-uniex.1 . 2  |-  A  e. 
_V
2 unieq 3745 . . 3  |-  ( x  =  A  ->  U. x  =  U. A )
32eleq1d 2208 . 2  |-  ( x  =  A  ->  ( U. x  e.  _V  <->  U. A  e.  _V )
)
4 bj-uniex2 13144 . . 3  |-  E. y 
y  =  U. x
54issetri 2695 . 2  |-  U. x  e.  _V
61, 3, 5vtocl 2740 1  |-  U. A  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480   _Vcvv 2686   U.cuni 3736
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-un 4355  ax-bd0 13041  ax-bdex 13047  ax-bdel 13049  ax-bdsb 13050  ax-bdsep 13112
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-uni 3737  df-bdc 13069
This theorem is referenced by:  bj-uniexg  13146  bj-unex  13147
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