Theorem uniexg 4530

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 3897	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2298	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2802	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4528	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2861	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ∪ cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-uni 3889

This theorem is referenced by:  uniexd  4531  abnexg  4537  snnex  4539  uniexb  4564  ssonuni  4580  dmexg  4988  rnexg  4989  elxp4  5216  elxp5  5217  iotaexab  5297  relrnfvex  5647  fvexg  5648  sefvex  5650  riotaexg  5964  iunexg  6270  1stvalg  6294  2ndvalg  6295  cnvf1o  6377  brtpos2  6403  tfrlemiex  6483  tfr1onlemex  6499  tfrcllemex  6512  en1bg  6960  en1uniel  6964  fival  7148  suplocexprlem2b  7912  suplocexprlemlub  7922  wrdexb  11096  restid  13299  tgval  13311  tgvalex  13312  istopon  14703  eltg  14742  eltg2  14743  tgss2  14769  ntrval  14800  restin  14866  cnovex  14886  cnprcl2k  14896  cnptopresti  14928  cnptoprest  14929  cnptoprest2  14930  lmtopcnp  14940  txbasex  14947  uptx  14964  reldvg  15369