Theorem uniexg 4201

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 3618	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2148	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2605	. . 3 |- x e. _V
4	3	uniex 4200	. 2 |- U. x e. _V
5	2, 4	vtoclg 2659	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   _Vcvv 2602   U.cuni 3609

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-un 4196

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-uni 3610

This theorem is referenced by:  snnex  4207  uniexb  4231  ssonuni  4240  dmexg  4624  rnexg  4625  elxp4  4838  elxp5  4839  relrnfvex  5224  fvexg  5225  sefvex  5227  riotaexg  5503  iunexg  5777  1stvalg  5800  2ndvalg  5801  cnvf1o  5877  brtpos2  5900  tfrlemiex  5980  tfr1onlemex  5996  tfrcllemex  6009  en1bg  6347  en1uniel  6351