Theorem uniexg 4560

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 3923	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2301	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2816	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4558	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2875	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  Vcvv 2813  ∪ cuni 3914

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-uni 3915

This theorem is referenced by:  uniexd  4561  abnexg  4567  snnex  4569  uniexb  4594  ssonuni  4610  dmexg  5021  rnexg  5022  elxp4  5250  elxp5  5251  iotaexab  5331  relrnfvex  5688  fvexg  5689  sefvex  5691  riotaexg  6007  iunexg  6312  1stvalg  6336  2ndvalg  6337  cnvf1o  6421  brtpos2  6482  tfrlemiex  6562  tfr1onlemex  6578  tfrcllemex  6591  en1bg  7040  en1uniel  7044  fival  7257  suplocexprlem2b  8029  suplocexprlemlub  8039  wrdexb  11236  restid  13463  tgval  13475  tgvalex  13476  istopon  14878  eltg  14917  eltg2  14918  tgss2  14944  ntrval  14975  restin  15041  cnovex  15061  cnprcl2k  15071  cnptopresti  15103  cnptoprest  15104  cnptoprest2  15105  lmtopcnp  15115  txbasex  15122  uptx  15139  reldvg  15544