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Theorem dvdszrcl 11509
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 11505 . . 3 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2 opabssxp 4613 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} ⊆ (ℤ × ℤ)
31, 2eqsstri 3129 . 2 ∥ ⊆ (ℤ × ℤ)
43brel 4591 1 (𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  wrex 2417   class class class wbr 3929  {copab 3988   × cxp 4537  (class class class)co 5774   · cmul 7637  cz 9066  cdvds 11504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-dvds 11505
This theorem is referenced by:  dvdsabseq  11556  divconjdvds  11558  evenelz  11575  4dvdseven  11625  dfgcd2  11713  dvdsmulgcd  11724  isprm3  11810  dvdsnprmd  11817
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