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  5645  fvexg  5646  sefvex  5648  riotaexg  5958  iunexg  6264  1stvalg  6288  2ndvalg  6289  cnvf1o  6371  brtpos2  6397  tfrlemiex  6477  tfr1onlemex  6493  tfrcllemex  6506  en1bg  6952  en1uniel  6956  fival  7137  suplocexprlem2b  7901  suplocexprlemlub  7911  wrdexb  11083  restid  13283  tgval  13295  tgvalex  13296  istopon  14687  eltg  14726  eltg2  14727  tgss2  14753  ntrval  14784  restin  14850  cnovex  14870  cnprcl2k  14880  cnptopresti  14912  cnptoprest  14913  cnptoprest2  14914  lmtopcnp  14924  txbasex  14931  uptx  14948  reldvg  15353