Theorem uniexg 4585

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 3944	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2307	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2824	. . 3 |- x e. _V
4	3	uniex 4583	. 2 |- U. x e. _V
5	2, 4	vtoclg 2883	1 |- ( A e. V -> U. A e. _V )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-un 4578

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-uni 3936

This theorem is used by:  uniexd  4586  abnexg  4592  snnex  4594  uniexb  4619  ssonuni  4635  dmexg  5046  rnexg  5047  elxp4  5275  elxp5  5276  iotaexab  5356  relrnfvex  5713  fvexg  5714  sefvex  5716  riotaexg  6042  iunexg  6348  1stvalg  6376  2ndvalg  6377  cnvf1o  6461  brtpos2  6522  tfrlemiex  6602  tfr1onlemex  6618  tfrcllemex  6631  en1bg  7087  en1uniel  7091  fival  7304  suplocexprlem2b  8081  suplocexprlemlub  8091  wrdexb  11316  restid  13604  tgval  13616  tgvalex  13617  istopon  15114  eltg  15153  eltg2  15154  tgss2  15180  ntrval  15211  restin  15277  cnovex  15297  cnprcl2k  15307  cnptopresti  15339  cnptoprest  15340  cnptoprest2  15341  lmtopcnp  15351  txbasex  15358  uptx  15375  reldvg  15780