Users' Mathboxes Mathbox for Steve Rodriguez < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reldvds Structured version   Visualization version   GIF version

Theorem reldvds 40870
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 15599 . 2 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
21relopabi 5677 1 Rel ∥
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  wrex 3133  Rel wrel 5543  (class class class)co 7140   · cmul 10529  cz 11969  cdvds 15598
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-opab 5112  df-xp 5544  df-rel 5545  df-dvds 15599
This theorem is referenced by:  nznngen  40871  nzss  40872  nzin  40873  hashnzfz  40875
  Copyright terms: Public domain W3C validator