Theorem reldvds 42306

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 16063	. 2 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2	1	relopabiv 5766	1 ⊢ Rel ∥

Syntax hints:   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ∃wrex 3071  Rel wrel 5629  (class class class)co 7341   · cmul 10981  ℤcz 12424   ∥ cdvds 16062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3444  df-in 3908  df-ss 3918  df-opab 5159  df-xp 5630  df-rel 5631  df-dvds 16063

This theorem is referenced by:  nznngen  42307  nzss  42308  nzin  42309  hashnzfz  42311