Theorem uniex 4472

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2769), 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 3848	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2265	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4471	. . 3 |- E. y y = U. x
5	4	issetri 2772	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2818	1 |- U. A e. _V

Syntax hints:   = wceq 1364   e. wcel 2167   _Vcvv 2763   U.cuni 3839

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-uni 3840

This theorem is referenced by:  vuniex  4473  uniexg  4474  unex  4476  uniuni  4486  iunpw  4515  fo1st  6215  fo2nd  6216  brtpos2  6309  tfrexlem  6392  ixpsnf1o  6795  xpcomco  6885  xpassen  6889  pnfnre  8068  pnfxr  8079