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Theorem drsprs 15889
Description: A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
drsprs  |-  ( K  e. Dirset  ->  K  e.  Preset  )

Proof of Theorem drsprs
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2402 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 15887 . 2  |-  ( K  e. Dirset 
<->  ( K  e.  Preset  /\  ( Base `  K
)  =/=  (/)  /\  A. x  e.  ( Base `  K ) A. y  e.  ( Base `  K
) E. z  e.  ( Base `  K
) ( x ( le `  K ) z  /\  y ( le `  K ) z ) ) )
43simp1bi 1012 1  |-  ( K  e. Dirset  ->  K  e.  Preset  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   (/)c0 3738   class class class wbr 4395   ` cfv 5569   Basecbs 14841   lecple 14916    Preset cpreset 15879  Dirsetcdrs 15880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-drs 15882
This theorem is referenced by:  drsdirfi  15891  isdrs2  15892
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