Theorem uniex 4264

Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2625), 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 3662	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2156	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		uniex2 4263	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2628	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2673	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1289   ∈ wcel 1438  Vcvv 2619  ∪ cuni 3653

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-uni 3654

This theorem is referenced by:  uniexg  4265  unex  4266  uniuni  4273  iunpw  4302  fo1st  5928  fo2nd  5929  brtpos2  6016  tfrexlem  6099  xpcomco  6542  xpassen  6546  pnfnre  7529  pnfxr  7540