Theorem uniexg 4193

Description: The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent A e. V instead of A e. _V to make the theorem more general and thus shorten some proofs; obviously the universal class constant _V is one possible substitution for class variable V. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
uniexg	|- ( A e. V -> U. A e. _V )

Proof of Theorem uniexg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 3610	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2147	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2604	. . 3 |- x e. _V
4	3	uniex 4192	. 2 |- U. x e. _V
5	2, 4	vtoclg 2658	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433   _Vcvv 2601   U.cuni 3601

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-uni 3602

This theorem is referenced by:  snnex  4199  uniexb  4223  ssonuni  4232  dmexg  4614  rnexg  4615  elxp4  4828  elxp5  4829  relrnfvex  5213  fvexg  5214  sefvex  5216  riotaexg  5492  iunexg  5766  1stvalg  5789  2ndvalg  5790  cnvf1o  5866  brtpos2  5889  tfrlemiex  5968  en1bg  6303  en1uniel  6307