Theorem uniexg 4474

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 3848	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2265	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2766	. . 3 |- x e. _V
4	3	uniex 4472	. 2 |- U. x e. _V
5	2, 4	vtoclg 2824	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   _Vcvv 2763   U.cuni 3839

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-uni 3840

This theorem is referenced by:  uniexd  4475  abnexg  4481  snnex  4483  uniexb  4508  ssonuni  4524  dmexg  4930  rnexg  4931  elxp4  5157  elxp5  5158  iotaexab  5237  relrnfvex  5576  fvexg  5577  sefvex  5579  riotaexg  5881  iunexg  6176  1stvalg  6200  2ndvalg  6201  cnvf1o  6283  brtpos2  6309  tfrlemiex  6389  tfr1onlemex  6405  tfrcllemex  6418  en1bg  6859  en1uniel  6863  fival  7036  suplocexprlem2b  7781  suplocexprlemlub  7791  wrdexb  10947  restid  12921  tgval  12933  tgvalex  12934  istopon  14249  eltg  14288  eltg2  14289  tgss2  14315  ntrval  14346  restin  14412  cnovex  14432  cnprcl2k  14442  cnptopresti  14474  cnptoprest  14475  cnptoprest2  14476  lmtopcnp  14486  txbasex  14493  uptx  14510  reldvg  14915