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Theorem uniex 4124
Description: The Axiom of Union in class notation. This says that if A is a set i.e. A V (see isset 2539), 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 V
Assertion
Ref Expression
uniex A 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 V
2 unieq 3563 . . 3 (x = A x = A)
32eleq1d 2088 . 2 (x = A → ( x V ↔ A V))
4 uniex2 4123 . . 3 y y = x
54issetri 2542 . 2 x V
61, 3, 5vtocl 2585 1 A V
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  Vcvv 2535   cuni 3554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-uni 3555
This theorem is referenced by:  uniexg  4125  unex  4126  uniuni  4133  iunpw  4161  fo1st  5707  fo2nd  5708  brtpos2  5788  tfrexlem  5870  pnfnre  6668
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