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Theorem bj-uniex 13912
Description: uniex 4420 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 3803 . . 3  |-  ( x  =  A  ->  U. x  =  U. A )
32eleq1d 2239 . 2  |-  ( x  =  A  ->  ( U. x  e.  _V  <->  U. A  e.  _V )
)
4 bj-uniex2 13911 . . 3  |-  E. y 
y  =  U. x
54issetri 2739 . 2  |-  U. x  e.  _V
61, 3, 5vtocl 2784 1  |-  U. A  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141   _Vcvv 2730   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-un 4416  ax-bd0 13808  ax-bdex 13814  ax-bdel 13816  ax-bdsb 13817  ax-bdsep 13879
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-uni 3795  df-bdc 13836
This theorem is referenced by:  bj-uniexg  13913  bj-unex  13914
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