Theorem uniexg 4475

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 3849	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2265	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2766	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4473	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2824	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  Vcvv 2763  ∪ cuni 3840

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-uni 3841

This theorem is referenced by:  uniexd  4476  abnexg  4482  snnex  4484  uniexb  4509  ssonuni  4525  dmexg  4931  rnexg  4932  elxp4  5158  elxp5  5159  iotaexab  5238  relrnfvex  5577  fvexg  5578  sefvex  5580  riotaexg  5882  iunexg  6177  1stvalg  6201  2ndvalg  6202  cnvf1o  6284  brtpos2  6310  tfrlemiex  6390  tfr1onlemex  6406  tfrcllemex  6419  en1bg  6860  en1uniel  6864  fival  7037  suplocexprlem2b  7783  suplocexprlemlub  7793  wrdexb  10949  restid  12931  tgval  12943  tgvalex  12944  istopon  14259  eltg  14298  eltg2  14299  tgss2  14325  ntrval  14356  restin  14422  cnovex  14442  cnprcl2k  14452  cnptopresti  14484  cnptoprest  14485  cnptoprest2  14486  lmtopcnp  14496  txbasex  14503  uptx  14520  reldvg  14925