Theorem uniexg 4587

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 3944	. . 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 4585	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2883	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ∪ 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 4251  ax-un 4580

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  4588  abnexg  4594  snnex  4596  uniexb  4621  ssonuni  4637  dmexg  5048  rnexg  5049  elxp4  5278  elxp5  5279  iotaexab  5359  relrnfvex  5716  fvexg  5717  sefvex  5719  riotaexg  6045  iunexg  6352  1stvalg  6380  2ndvalg  6381  cnvf1o  6465  brtpos2  6527  tfrlemiex  6607  tfr1onlemex  6623  tfrcllemex  6636  en1bg  7093  en1uniel  7097  fival  7309  suplocexprlem2b  8087  suplocexprlemlub  8097  wrdexb  11337  restid  13662  tgval  13674  tgvalex  13675  istopon  15256  eltg  15295  eltg2  15296  tgss2  15322  ntrval  15353  restin  15419  cnovex  15439  cnprcl2k  15449  cnptopresti  15481  cnptoprest  15482  cnptoprest2  15483  lmtopcnp  15493  txbasex  15500  uptx  15517  reldvg  15922