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 37413
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 14686 . 2 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
21relopabi 5060 1 Rel ∥
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1938  wrex 2801  Rel wrel 4937  (class class class)co 6425   · cmul 9694  cz 11116  cdvds 14685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-opab 4542  df-xp 4938  df-rel 4939  df-dvds 14686
This theorem is referenced by:  nznngen  37414  nzss  37415  nzin  37416  hashnzfz  37418
  Copyright terms: Public domain W3C validator