Theorem uniex 4484

Description: The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2778), 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 3859	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
3	2	eleq1d 2274	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
4		uniex2 4483	. . 3 ⊢ ∃𝑦 𝑦 = ∪ 𝑥
5	4	issetri 2781	. 2 ⊢ ∪ 𝑥 ∈ V
6	1, 3, 5	vtocl 2827	1 ⊢ ∪ 𝐴 ∈ V

Syntax hints:   = wceq 1373   ∈ wcel 2176  Vcvv 2772  ∪ cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-uni 3851

This theorem is referenced by:  vuniex  4485  uniexg  4486  unex  4488  uniuni  4498  iunpw  4527  fo1st  6243  fo2nd  6244  brtpos2  6337  tfrexlem  6420  ixpsnf1o  6823  xpcomco  6921  xpassen  6925  pnfnre  8114  pnfxr  8125  prdsvallem  13104  prdsval  13105