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Theorem uniex 4140
Description: The Axiom of Union in class notation. This says that if A is a set i.e. A V (see isset 2555), 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 3580 . . 3 (x = A x = A)
32eleq1d 2103 . 2 (x = A → ( x V ↔ A V))
4 uniex2 4139 . . 3 y y = x
54issetri 2558 . 2 x V
61, 3, 5vtocl 2602 1 A V
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551   cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-uni 3572
This theorem is referenced by:  uniexg  4141  unex  4142  uniuni  4149  iunpw  4177  fo1st  5726  fo2nd  5727  brtpos2  5807  tfrexlem  5889  xpcomco  6236  xpassen  6240  pnfnre  6824  pnfxr  8422
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