Theorem uniexg 4504

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 3873	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2276	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2779	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4502	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2838	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178  Vcvv 2776  ∪ cuni 3864

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-uni 3865

This theorem is referenced by:  uniexd  4505  abnexg  4511  snnex  4513  uniexb  4538  ssonuni  4554  dmexg  4961  rnexg  4962  elxp4  5189  elxp5  5190  iotaexab  5269  relrnfvex  5617  fvexg  5618  sefvex  5620  riotaexg  5926  iunexg  6227  1stvalg  6251  2ndvalg  6252  cnvf1o  6334  brtpos2  6360  tfrlemiex  6440  tfr1onlemex  6456  tfrcllemex  6469  en1bg  6915  en1uniel  6919  fival  7098  suplocexprlem2b  7862  suplocexprlemlub  7872  wrdexb  11043  restid  13197  tgval  13209  tgvalex  13210  istopon  14600  eltg  14639  eltg2  14640  tgss2  14666  ntrval  14697  restin  14763  cnovex  14783  cnprcl2k  14793  cnptopresti  14825  cnptoprest  14826  cnptoprest2  14827  lmtopcnp  14837  txbasex  14844  uptx  14861  reldvg  15266