Theorem reldvds 41933

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 15964	. 2 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2	1	relopabiv 5730	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Rel wrel 5594  (class class class)co 7275   · cmul 10876  ℤcz 12319   ∥ cdvds 15963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596  df-dvds 15964

This theorem is referenced by:  nznngen  41934  nzss  41935  nzin  41936  hashnzfz  41938