Theorem uniex 4194

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2606), 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 e. _V

Assertion

Ref	Expression
uniex	|- U. A e. _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 e. _V
2		unieq 3612	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2148	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4193	. . 3 |- E. y y = U. x
5	4	issetri 2609	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2654	1 |- U. A e. _V

Syntax hints:   = wceq 1285   e. wcel 1434   _Vcvv 2602   U.cuni 3603

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-un 4190

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-uni 3604

This theorem is referenced by:  uniexg  4195  unex  4196  uniuni  4203  iunpw  4231  fo1st  5809  fo2nd  5810  brtpos2  5894  tfrexlem  5977  xpcomco  6360  xpassen  6364  pnfnre  7211  pnfxr  7222