Theorem uniex 4438

Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2744), 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 3819	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2246	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		uniex2 4437	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2747	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2792	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1353   ∈ wcel 2148  Vcvv 2738  ∪ cuni 3810

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-uni 3811

This theorem is referenced by:  vuniex  4439  uniexg  4440  unex  4442  uniuni  4452  iunpw  4481  fo1st  6158  fo2nd  6159  brtpos2  6252  tfrexlem  6335  ixpsnf1o  6736  xpcomco  6826  xpassen  6830  pnfnre  7999  pnfxr  8010