Theorem uniexg 4471

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 3845	. . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
2	1	eleq1d 2262	. 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ V ↔ ∪ 𝐴 ∈ V))
3		vex 2763	. . 3 ⊢ 𝑥 ∈ V
4	3	uniex 4469	. 2 ⊢ ∪ 𝑥 ∈ V
5	2, 4	vtoclg 2821	1 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760  ∪ cuni 3836

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-uni 3837

This theorem is referenced by:  uniexd  4472  abnexg  4478  snnex  4480  uniexb  4505  ssonuni  4521  dmexg  4927  rnexg  4928  elxp4  5154  elxp5  5155  iotaexab  5234  relrnfvex  5573  fvexg  5574  sefvex  5576  riotaexg  5878  iunexg  6173  1stvalg  6197  2ndvalg  6198  cnvf1o  6280  brtpos2  6306  tfrlemiex  6386  tfr1onlemex  6402  tfrcllemex  6415  en1bg  6856  en1uniel  6860  fival  7031  suplocexprlem2b  7776  suplocexprlemlub  7786  wrdexb  10929  restid  12864  tgval  12876  tgvalex  12877  istopon  14192  eltg  14231  eltg2  14232  tgss2  14258  ntrval  14289  restin  14355  cnovex  14375  cnprcl2k  14385  cnptopresti  14417  cnptoprest  14418  cnptoprest2  14419  lmtopcnp  14429  txbasex  14436  uptx  14453  reldvg  14858