Theorem reldvds 44347

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 16161	. 2 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2	1	relopabiv 5760	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Rel wrel 5621  (class class class)co 7346   · cmul 11008  ℤcz 12465   ∥ cdvds 16160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3919  df-opab 5154  df-xp 5622  df-rel 5623  df-dvds 16161

This theorem is referenced by:  nznngen  44348  nzss  44349  nzin  44350  hashnzfz  44352