Theorem uniexg 4440

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 3819	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2246	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2741	. . 3 |- x e. _V
4	3	uniex 4438	. 2 |- U. x e. _V
5	2, 4	vtoclg 2798	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2738   U.cuni 3810

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-uni 3811

This theorem is referenced by:  uniexd  4441  abnexg  4447  snnex  4449  uniexb  4474  ssonuni  4488  dmexg  4892  rnexg  4893  elxp4  5117  elxp5  5118  relrnfvex  5534  fvexg  5535  sefvex  5537  riotaexg  5835  iunexg  6120  1stvalg  6143  2ndvalg  6144  cnvf1o  6226  brtpos2  6252  tfrlemiex  6332  tfr1onlemex  6348  tfrcllemex  6361  en1bg  6800  en1uniel  6804  fival  6969  suplocexprlem2b  7713  suplocexprlemlub  7723  restid  12699  tgval  12711  tgvalex  12712  istopon  13516  eltg  13555  eltg2  13556  tgss2  13582  ntrval  13613  restin  13679  cnovex  13699  cnprcl2k  13709  cnptopresti  13741  cnptoprest  13742  cnptoprest2  13743  lmtopcnp  13753  txbasex  13760  uptx  13777  reldvg  14151