Theorem uniex 4583

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 3944	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2307	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		uniex2 4581	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2831	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2877	1 ⊢ ∪ 𝐴 ∈ V

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∪ cuni 3935

This proof depends on 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 4249  ax-un 4578

This proof 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 3936

This theorem is used by:  vuniex  4584  uniexg  4585  unex  4587  uniuni  4597  iunpw  4626  fo1st  6391  fo2nd  6392  brtpos2  6522  tfrexlem  6605  ixpsnf1o  7018  xpcomco  7124  xpassen  7128  pnfnre  8367  pnfxr  8378  prdsvallem  13621  prdsval  14173