Theorem uniex 4536

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2808), 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 3903	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2299	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4535	. . 3 |- E. y y = U. x
5	4	issetri 2811	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2857	1 |- U. A e. _V

Syntax hints:   = wceq 1397   e. wcel 2201   _Vcvv 2801   U.cuni 3894

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-v 2803  df-uni 3895

This theorem is referenced by:  vuniex  4537  uniexg  4538  unex  4540  uniuni  4550  iunpw  4579  fo1st  6325  fo2nd  6326  brtpos2  6422  tfrexlem  6505  ixpsnf1o  6910  xpcomco  7015  xpassen  7019  pnfnre  8226  pnfxr  8237  prdsvallem  13378  prdsval  13379