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Theorem dvdszrcl 14002
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 13998 . . 3  |-  ||  =  { <. x ,  y
>.  |  ( (
x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x
)  =  y ) }
2 opabssxp 5083 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  ZZ  /\  y  e.  ZZ )  /\  E. z  e.  ZZ  ( z  x.  x )  =  y ) }  C_  ( ZZ  X.  ZZ )
31, 2eqsstri 3529 . 2  |-  ||  C_  ( ZZ  X.  ZZ )
43brel 5057 1  |-  ( X 
||  Y  ->  ( X  e.  ZZ  /\  Y  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   class class class wbr 4456   {copab 4514    X. cxp 5006  (class class class)co 6296    x. cmul 9514   ZZcz 10885    || cdvds 13997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-dvds 13998
This theorem is referenced by:  dvdsmulgcd  14203  oddvdsi  16698  odmulg  16704  gexdvdsi  16729  numclwwlk8  25241  nzss  31384  nzin  31385
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