Theorem uniexg 4399

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 3781	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2226	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2715	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4397	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2772	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  Vcvv 2712  ∪ cuni 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-uni 3773

This theorem is referenced by:  uniexd  4400  abnexg  4406  snnex  4408  uniexb  4433  ssonuni  4447  dmexg  4850  rnexg  4851  elxp4  5073  elxp5  5074  relrnfvex  5486  fvexg  5487  sefvex  5489  riotaexg  5784  iunexg  6067  1stvalg  6090  2ndvalg  6091  cnvf1o  6172  brtpos2  6198  tfrlemiex  6278  tfr1onlemex  6294  tfrcllemex  6307  en1bg  6745  en1uniel  6749  fival  6914  suplocexprlem2b  7634  suplocexprlemlub  7644  restid  12373  istopon  12422  tgval  12460  tgvalex  12461  eltg  12463  eltg2  12464  tgss2  12490  ntrval  12521  restin  12587  cnovex  12607  cnprcl2k  12617  cnptopresti  12649  cnptoprest  12650  cnptoprest2  12651  lmtopcnp  12661  txbasex  12668  uptx  12685  reldvg  13059