Theorem reldvds 44951

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 16311	. 2 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2	1	relopabiv 5808	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Rel wrel 5667  (class class class)co 7411   · cmul 11105  ℤcz 12591   ∥ cdvds 16310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-opab 5178  df-xp 5668  df-rel 5669  df-dvds 16311

This theorem is referenced by:  nznngen  44952  nzss  44953  nzin  44954  hashnzfz  44956