Theorem uniex 4439

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2745), 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 3820	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2246	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4438	. . 3 |- E. y y = U. x
5	4	issetri 2748	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2793	1 |- U. A e. _V

Syntax hints:   = wceq 1353   e. wcel 2148   _Vcvv 2739   U.cuni 3811

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-uni 3812

This theorem is referenced by:  vuniex  4440  uniexg  4441  unex  4443  uniuni  4453  iunpw  4482  fo1st  6160  fo2nd  6161  brtpos2  6254  tfrexlem  6337  ixpsnf1o  6738  xpcomco  6828  xpassen  6832  pnfnre  8001  pnfxr  8012