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Theorem uniex 4201
 Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2578), 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 3616 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
32eleq1d 2122 . 2 (𝑥 = 𝐴 → ( 𝑥 ∈ V ↔ 𝐴 ∈ V))
4 uniex2 4200 . . 3 𝑦 𝑦 = 𝑥
54issetri 2581 . 2 𝑥 ∈ V
61, 3, 5vtocl 2625 1 𝐴 ∈ V
 Colors of variables: wff set class Syntax hints:   = wceq 1259   ∈ wcel 1409  Vcvv 2574  ∪ cuni 3607 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-un 4197 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-uni 3608 This theorem is referenced by:  uniexg  4202  unex  4203  uniuni  4210  iunpw  4238  fo1st  5811  fo2nd  5812  brtpos2  5896  tfrexlem  5978  xpcomco  6330  xpassen  6334  pnfnre  7125  pnfxr  8792
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