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Theorem drsprs 14313
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 2380 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2380 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 14311 . 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 1717    =/= wne 2543   A.wral 2642   E.wrex 2643   (/)c0 3564   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456    Preset cpreset 14303  Dirsetcdrs 14304
This theorem is referenced by:  drsdirfi  14315  isdrs2  14316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-nul 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-drs 14306
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