Theorem uniex 4560

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2822), 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 3925	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2303	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4559	. . 3 |- E. y y = U. x
5	4	issetri 2825	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2871	1 |- U. A e. _V

Syntax hints:   = wceq 1398   e. wcel 2205   _Vcvv 2815   U.cuni 3916

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-uni 3917

This theorem is referenced by:  vuniex  4561  uniexg  4562  unex  4564  uniuni  4574  iunpw  4603  fo1st  6353  fo2nd  6354  brtpos2  6484  tfrexlem  6567  ixpsnf1o  6973  xpcomco  7079  xpassen  7083  pnfnre  8320  pnfxr  8331  prdsvallem  13506  prdsval  13507