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Theorem uniexg 4256
Description: The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent  A  e.  V instead of  A  e.  _V to make the theorem more general and thus shorten some proofs; obviously the universal class constant  _V is one possible substitution for class variable  V. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
uniexg  |-  ( A  e.  V  ->  U. A  e.  _V )

Proof of Theorem uniexg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 unieq 3657 . . 3  |-  ( x  =  A  ->  U. x  =  U. A )
21eleq1d 2156 . 2  |-  ( x  =  A  ->  ( U. x  e.  _V  <->  U. A  e.  _V )
)
3 vex 2622 . . 3  |-  x  e. 
_V
43uniex 4255 . 2  |-  U. x  e.  _V
52, 4vtoclg 2679 1  |-  ( A  e.  V  ->  U. A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   U.cuni 3648
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-uni 3649
This theorem is referenced by:  snnex  4262  uniexb  4286  ssonuni  4295  dmexg  4685  rnexg  4686  elxp4  4905  elxp5  4906  relrnfvex  5307  fvexg  5308  sefvex  5310  riotaexg  5594  iunexg  5872  1stvalg  5895  2ndvalg  5896  cnvf1o  5972  brtpos2  5998  tfrlemiex  6078  tfr1onlemex  6094  tfrcllemex  6107  en1bg  6497  en1uniel  6501
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