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Theorem dvdszrcl 16295
Description: Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Assertion
Ref Expression
dvdszrcl (𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))

Proof of Theorem dvdszrcl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvds 16291 . . 3 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2 opabssxp 5778 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} ⊆ (ℤ × ℤ)
31, 2eqsstri 4030 . 2 ∥ ⊆ (ℤ × ℤ)
43brel 5750 1 (𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070   class class class wbr 5143  {copab 5205   × cxp 5683  (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  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-dvds 16291
This theorem is referenced by:  dvdsmod0  16296  p1modz1  16297  dvdsmodexp  16298  dvdsaddre2b  16344  dvdsabseq  16350  divconjdvds  16352  evenelz  16373  4dvdseven  16410  dfgcd2  16583  dvdsmulgcd  16593  dvdsnprmd  16727  dvdszzq  16758  oddvdsi  19566  odmulg  19574  gexdvdsi  19601  dvdschrmulg  21543  nnproddivdvdsd  42001  lcmineqlem14  42043  aks6d1c6isolem3  42177  grpods  42195  nzss  44336  nzin  44337
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