Theorem uniex 4532

Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2807), 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 3900	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2298	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		uniex2 4531	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2810	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2856	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2800  ∪ cuni 3891

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-uni 3892

This theorem is referenced by:  vuniex  4533  uniexg  4534  unex  4536  uniuni  4546  iunpw  4575  fo1st  6315  fo2nd  6316  brtpos2  6412  tfrexlem  6495  ixpsnf1o  6900  xpcomco  7005  xpassen  7009  pnfnre  8211  pnfxr  8222  prdsvallem  13345  prdsval  13346