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Theorem drsprs 14386
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 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2436 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 14384 . 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 972 1  |-  ( K  e. Dirset  ->  K  e.  Preset  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    =/= wne 2599   A.wral 2698   E.wrex 2699   (/)c0 3621   class class class wbr 4205   ` cfv 5447   Basecbs 13462   lecple 13529    Preset cpreset 14376  Dirsetcdrs 14377
This theorem is referenced by:  drsdirfi  14388  isdrs2  14389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4331
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-iota 5411  df-fv 5455  df-drs 14379
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