Theorem uniex 4558

Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2820), 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 3923	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2301	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		uniex2 4557	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2823	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2869	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1398   ∈ wcel 2203  Vcvv 2813  ∪ cuni 3914

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-uni 3915

This theorem is referenced by:  vuniex  4559  uniexg  4560  unex  4562  uniuni  4572  iunpw  4601  fo1st  6351  fo2nd  6352  brtpos2  6482  tfrexlem  6565  ixpsnf1o  6971  xpcomco  7077  xpassen  7081  pnfnre  8315  pnfxr  8326  prdsvallem  13485  prdsval  13486