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Theorem drsprs 14086
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 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2296 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 14084 . 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 970 1  |-  ( K  e. Dirset  ->  K  e.  Preset  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   (/)c0 3468   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231    Preset cpreset 14076  Dirsetcdrs 14077
This theorem is referenced by:  drsdirfi  14088  isdrs2  14089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-drs 14079
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