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Theorem uniex 4194
Description: The Axiom of Union in class notation. This says that if 
A is a set i.e.  A  e.  _V (see isset 2606), 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 3612 . . 3  |-  ( x  =  A  ->  U. x  =  U. A )
32eleq1d 2148 . 2  |-  ( x  =  A  ->  ( U. x  e.  _V  <->  U. A  e.  _V )
)
4 uniex2 4193 . . 3  |-  E. y 
y  =  U. x
54issetri 2609 . 2  |-  U. x  e.  _V
61, 3, 5vtocl 2654 1  |-  U. A  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   _Vcvv 2602   U.cuni 3603
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-un 4190
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-uni 3604
This theorem is referenced by:  uniexg  4195  unex  4196  uniuni  4203  iunpw  4231  fo1st  5809  fo2nd  5810  brtpos2  5894  tfrexlem  5977  xpcomco  6360  xpassen  6364  pnfnre  7211  pnfxr  7222
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