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Theorem uniex 4455
Description: The Axiom of Union in class notation. This says that if 
A is a set i.e.  A  e.  _V (see isset 2758), 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 3833 . . 3  |-  ( x  =  A  ->  U. x  =  U. A )
32eleq1d 2258 . 2  |-  ( x  =  A  ->  ( U. x  e.  _V  <->  U. A  e.  _V )
)
4 uniex2 4454 . . 3  |-  E. y 
y  =  U. x
54issetri 2761 . 2  |-  U. x  e.  _V
61, 3, 5vtocl 2806 1  |-  U. A  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   _Vcvv 2752   U.cuni 3824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-uni 3825
This theorem is referenced by:  vuniex  4456  uniexg  4457  unex  4459  uniuni  4469  iunpw  4498  fo1st  6183  fo2nd  6184  brtpos2  6277  tfrexlem  6360  ixpsnf1o  6763  xpcomco  6853  xpassen  6857  pnfnre  8030  pnfxr  8041
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