Theorem uniex 4192

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2605), 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 3610	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2147	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4191	. . 3 |- E. y y = U. x
5	4	issetri 2608	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2653	1 |- U. A e. _V

Syntax hints:   = wceq 1284   e. wcel 1433   _Vcvv 2601   U.cuni 3601

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-uni 3602

This theorem is referenced by:  uniexg  4193  unex  4194  uniuni  4201  iunpw  4229  fo1st  5804  fo2nd  5805  brtpos2  5889  tfrexlem  5971  xpcomco  6323  xpassen  6327  pnfnre  7160  pnfxr  8846