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Theorem dvdszrcl 14982
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 14978 . . 3 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2 opabssxp 5191 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} ⊆ (ℤ × ℤ)
31, 2eqsstri 3633 . 2 ∥ ⊆ (ℤ × ℤ)
43brel 5166 1 (𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wrex 2912   class class class wbr 4651  {copab 4710   × cxp 5110  (class class class)co 6647   · cmul 9938  cz 11374  cdvds 14977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-dvds 14978
This theorem is referenced by:  dvdsaddre2b  15023  dvdsabseq  15029  divconjdvds  15031  evenelz  15054  4dvdseven  15103  dfgcd2  15257  dvdsmulgcd  15268  dvdsnprmd  15397  oddvdsi  17961  odmulg  17967  gexdvdsi  17992  numclwwlk8  27234  nzss  38342  nzin  38343
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