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Theorem reldvds 44334
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.
StepHypRef Expression
1 df-dvds 16291 . 2 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
21relopabiv 5830 1 Rel ∥
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  wrex 3070  Rel wrel 5690  (class class class)co 7431   · cmul 11160  cz 12613  cdvds 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-opab 5206  df-xp 5691  df-rel 5692  df-dvds 16291
This theorem is referenced by:  nznngen  44335  nzss  44336  nzin  44337  hashnzfz  44339
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