Theorem uniex 4578

Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2828), 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.

Step	Hyp	Ref	Expression
1		uniex.1	. 2 ⊢ 𝐴 ∈ V
2		unieq 3939	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2307	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		uniex2 4576	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2831	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2877	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∪ cuni 3930

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-uni 3931

This theorem is referenced by:  vuniex  4579  uniexg  4580  unex  4582  uniuni  4592  iunpw  4621  fo1st  6381  fo2nd  6382  brtpos2  6512  tfrexlem  6595  ixpsnf1o  7008  xpcomco  7114  xpassen  7118  pnfnre  8357  pnfxr  8368  prdsvallem  13598  prdsval  14150