Theorem uniexg 4565

Description: The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴 ∈ 𝑉 instead of 𝐴 ∈ 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 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Proof of Theorem uniexg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 3928	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2303	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2818	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4563	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2877	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ∪ cuni 3919

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-uni 3920

This theorem is referenced by:  uniexd  4566  abnexg  4572  snnex  4574  uniexb  4599  ssonuni  4615  dmexg  5026  rnexg  5027  elxp4  5255  elxp5  5256  iotaexab  5336  relrnfvex  5693  fvexg  5694  sefvex  5696  riotaexg  6015  iunexg  6321  1stvalg  6349  2ndvalg  6350  cnvf1o  6434  brtpos2  6495  tfrlemiex  6575  tfr1onlemex  6591  tfrcllemex  6604  en1bg  7053  en1uniel  7057  fival  7270  suplocexprlem2b  8045  suplocexprlemlub  8055  wrdexb  11261  restid  13547  tgval  13559  tgvalex  13560  istopon  15004  eltg  15043  eltg2  15044  tgss2  15070  ntrval  15101  restin  15167  cnovex  15187  cnprcl2k  15197  cnptopresti  15229  cnptoprest  15230  cnptoprest2  15231  lmtopcnp  15241  txbasex  15248  uptx  15265  reldvg  15670