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Theorem dvdszrcl 12536
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 12532 . . 3  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x
)  =  y ) }
2 opabssxp 4762 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x )  =  y ) }  C_  ( ZZ  X.  ZZ )
31, 2eqsstri 3208 . 2  |-  ||  C_  ( ZZ  X.  ZZ )
43brel 4737 1  |-  ( X 
||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   {copab 4076    X. cxp 4687  (class class class)co 5858    x. cmul 8742   ZZcz 10024    || cdivides 12531
This theorem is referenced by:  dvdsmulgcd  12733  oddvdsi  14863  odmulg  14869  gexdvdsi  14894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dvds 12532
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