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Theorem uniex 4329
Description: The Axiom of Union in class notation. This says that if 
A is a set i.e.  A  e.  _V (see isset 2666), then the union of  A is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
Hypothesis
Ref Expression
uniex.1  |-  A  e. 
_V
Assertion
Ref Expression
uniex  |-  U. A  e.  _V

Proof of Theorem uniex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniex.1 . 2  |-  A  e. 
_V
2 unieq 3715 . . 3  |-  ( x  =  A  ->  U. x  =  U. A )
32eleq1d 2186 . 2  |-  ( x  =  A  ->  ( U. x  e.  _V  <->  U. A  e.  _V )
)
4 uniex2 4328 . . 3  |-  E. y 
y  =  U. x
54issetri 2669 . 2  |-  U. x  e.  _V
61, 3, 5vtocl 2714 1  |-  U. A  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   _Vcvv 2660   U.cuni 3706
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-uni 3707
This theorem is referenced by:  vuniex  4330  uniexg  4331  unex  4332  uniuni  4342  iunpw  4371  fo1st  6023  fo2nd  6024  brtpos2  6116  tfrexlem  6199  ixpsnf1o  6598  xpcomco  6688  xpassen  6692  pnfnre  7775  pnfxr  7786
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