Theorem uniexg 4580

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 3939	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2307	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2824	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4578	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2883	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∪ cuni 3930

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 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 4244  ax-un 4573

This theorem 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 3931

This theorem is referenced by:  uniexd  4581  abnexg  4587  snnex  4589  uniexb  4614  ssonuni  4630  dmexg  5041  rnexg  5042  elxp4  5270  elxp5  5271  iotaexab  5351  relrnfvex  5708  fvexg  5709  sefvex  5711  riotaexg  6032  iunexg  6338  1stvalg  6366  2ndvalg  6367  cnvf1o  6451  brtpos2  6512  tfrlemiex  6592  tfr1onlemex  6608  tfrcllemex  6621  en1bg  7077  en1uniel  7081  fival  7294  suplocexprlem2b  8071  suplocexprlemlub  8081  wrdexb  11294  restid  13581  tgval  13593  tgvalex  13594  istopon  15037  eltg  15076  eltg2  15077  tgss2  15103  ntrval  15134  restin  15200  cnovex  15220  cnprcl2k  15230  cnptopresti  15262  cnptoprest  15263  cnptoprest2  15264  lmtopcnp  15274  txbasex  15281  uptx  15298  reldvg  15703