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Theorem reldvds 44277
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 16278 . 2 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
21relopabiv 5828 1 Rel ∥
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1535  wcel 2104  wrex 3066  Rel wrel 5689  (class class class)co 7426   · cmul 11152  cz 12605  cdvds 16277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479  df-ss 3980  df-opab 5213  df-xp 5690  df-rel 5691  df-dvds 16278
This theorem is referenced by:  nznngen  44278  nzss  44279  nzin  44280  hashnzfz  44282
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