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Theorem dvdszrcl 16198
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 16194 . . 3 ∥ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)}
2 opabssxp 5766 . . 3 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑧 ∈ ℤ (𝑧 · 𝑥) = 𝑦)} ⊆ (ℤ × ℤ)
31, 2eqsstri 4015 . 2 ∥ ⊆ (ℤ × ℤ)
43brel 5739 1 (𝑋𝑌 → (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3071   class class class wbr 5147  {copab 5209   × cxp 5673  (class class class)co 7404   · cmul 11111  cz 12554  cdvds 16193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-dvds 16194
This theorem is referenced by:  dvdsmod0  16199  p1modz1  16200  dvdsmodexp  16201  dvdsaddre2b  16246  dvdsabseq  16252  divconjdvds  16254  evenelz  16275  4dvdseven  16312  dfgcd2  16484  dvdsmulgcd  16493  dvdsnprmd  16623  oddvdsi  19409  odmulg  19417  gexdvdsi  19444  dvdszzq  31999  dvdschrmulg  32355  nnproddivdvdsd  40804  lcmineqlem14  40845  nzss  43009  nzin  43010
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