Theorem uniex 4557

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2819), 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 3922	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2301	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4556	. . 3 |- E. y y = U. x
5	4	issetri 2822	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2868	1 |- U. A e. _V

Syntax hints:   = wceq 1398   e. wcel 2203   _Vcvv 2812   U.cuni 3913

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-uni 3914

This theorem is referenced by:  vuniex  4558  uniexg  4559  unex  4561  uniuni  4571  iunpw  4600  fo1st  6350  fo2nd  6351  brtpos2  6481  tfrexlem  6564  ixpsnf1o  6970  xpcomco  7076  xpassen  7080  pnfnre  8311  pnfxr  8322  prdsvallem  13474  prdsval  13475