Theorem uniex 4455

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2758), 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 3833	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2258	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4454	. . 3 |- E. y y = U. x
5	4	issetri 2761	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2806	1 |- U. A e. _V

Syntax hints:   = wceq 1364   e. wcel 2160   _Vcvv 2752   U.cuni 3824

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-uni 3825

This theorem is referenced by:  vuniex  4456  uniexg  4457  unex  4459  uniuni  4469  iunpw  4498  fo1st  6183  fo2nd  6184  brtpos2  6277  tfrexlem  6360  ixpsnf1o  6763  xpcomco  6853  xpassen  6857  pnfnre  8030  pnfxr  8041