Theorem uniex 4497

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2780), 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 3868	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2275	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4496	. . 3 |- E. y y = U. x
5	4	issetri 2783	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2829	1 |- U. A e. _V

Syntax hints:   = wceq 1373   e. wcel 2177   _Vcvv 2773   U.cuni 3859

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-uni 3860

This theorem is referenced by:  vuniex  4498  uniexg  4499  unex  4501  uniuni  4511  iunpw  4540  fo1st  6261  fo2nd  6262  brtpos2  6355  tfrexlem  6438  ixpsnf1o  6841  xpcomco  6941  xpassen  6945  pnfnre  8144  pnfxr  8155  prdsvallem  13189  prdsval  13190