Theorem uniexg 4542

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 3907	. . 3 |- ( x = A -> U. x = U. A )
2	1	eleq1d 2300	. 2 |- ( x = A -> ( U. x e. _V <-> U. A e. _V ) )
3		vex 2806	. . 3 |- x e. _V
4	3	uniex 4540	. 2 |- U. x e. _V
5	2, 4	vtoclg 2865	1 |- ( A e. V -> U. A e. _V )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   U.cuni 3898

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-uni 3899

This theorem is referenced by:  uniexd  4543  abnexg  4549  snnex  4551  uniexb  4576  ssonuni  4592  dmexg  5002  rnexg  5003  elxp4  5231  elxp5  5232  iotaexab  5312  relrnfvex  5666  fvexg  5667  sefvex  5669  riotaexg  5985  iunexg  6290  1stvalg  6314  2ndvalg  6315  cnvf1o  6399  brtpos2  6460  tfrlemiex  6540  tfr1onlemex  6556  tfrcllemex  6569  en1bg  7017  en1uniel  7021  fival  7229  suplocexprlem2b  7994  suplocexprlemlub  8004  wrdexb  11191  restid  13413  tgval  13425  tgvalex  13426  istopon  14824  eltg  14863  eltg2  14864  tgss2  14890  ntrval  14921  restin  14987  cnovex  15007  cnprcl2k  15017  cnptopresti  15049  cnptoprest  15050  cnptoprest2  15051  lmtopcnp  15061  txbasex  15068  uptx  15085  reldvg  15490