Theorem uniex 4237

Description: The Axiom of Union in class notation. This says that if A is a set i.e. A e. _V (see isset 2619), 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 3645	. . 3 |- ( x = A -> U. x = U. A )
3	2	eleq1d 2153	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
4		uniex2 4236	. . 3 |- E. y y = U. x
5	4	issetri 2622	. 2 |- U. x e. _V
6	1, 3, 5	vtocl 2667	1 |- U. A e. _V

Syntax hints:   = wceq 1287   e. wcel 1436   _Vcvv 2615   U.cuni 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-un 4233

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-uni 3637

This theorem is referenced by:  uniexg  4238  unex  4239  uniuni  4246  iunpw  4274  fo1st  5879  fo2nd  5880  brtpos2  5964  tfrexlem  6047  xpcomco  6488  xpassen  6492  pnfnre  7466  pnfxr  7477