Theorem reldvds 43817

Description: The divides relation is in fact a relation. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
reldvds	⊢ Rel ∥

Proof of Theorem reldvds

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvds 16231	. 2 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2	1	relopabiv 5821	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3060  Rel wrel 5682  (class class class)co 7417   · cmul 11143  ℤcz 12588   ∥ cdvds 16230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-ss 3962  df-opab 5211  df-xp 5683  df-rel 5684  df-dvds 16231

This theorem is referenced by:  nznngen  43818  nzss  43819  nzin  43820  hashnzfz  43822