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

Theorem reldvds 39034
 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 15203 . 2 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
21relopabi 5401 1 Rel ∥
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  Rel wrel 5271  (class class class)co 6814   · cmul 10153  ℤcz 11589   ∥ cdvds 15202 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273  df-dvds 15203 This theorem is referenced by:  nznngen  39035  nzss  39036  nzin  39037  hashnzfz  39039
 Copyright terms: Public domain W3C validator