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Theorem dvdszrcl 13539
Description: Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Assertion
Ref Expression
dvdszrcl  |-  ( X 
||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )

Proof of Theorem dvdszrcl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvds 13535 . . 3  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x
)  =  y ) }
2 opabssxp 4910 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x )  =  y ) }  C_  ( ZZ  X.  ZZ )
31, 2eqsstri 3385 . 2  |-  ||  C_  ( ZZ  X.  ZZ )
43brel 4886 1  |-  ( X 
||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2715   class class class wbr 4291   {copab 4348    X. cxp 4837  (class class class)co 6090    x. cmul 9286   ZZcz 10645    || cdivides 13534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-xp 4845  df-dvds 13535
This theorem is referenced by:  dvdsmulgcd  13737  oddvdsi  16050  odmulg  16056  gexdvdsi  16081  numclwwlk8  30706
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