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Theorem uniex 4354
Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2687), 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 3740 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
32eleq1d 2206 . 2 (𝑥 = 𝐴 → ( 𝑥 ∈ V ↔ 𝐴 ∈ V))
4 uniex2 4353 . . 3 𝑦 𝑦 = 𝑥
54issetri 2690 . 2 𝑥 ∈ V
61, 3, 5vtocl 2735 1 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  Vcvv 2681   cuni 3731
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-uni 3732
This theorem is referenced by:  vuniex  4355  uniexg  4356  unex  4357  uniuni  4367  iunpw  4396  fo1st  6048  fo2nd  6049  brtpos2  6141  tfrexlem  6224  ixpsnf1o  6623  xpcomco  6713  xpassen  6717  pnfnre  7800  pnfxr  7811
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