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Theorem drsprs 14070
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 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 14068 . 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 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215    Preset cpreset 14060  Dirsetcdrs 14061
This theorem is referenced by:  drsdirfi  14072  isdrs2  14073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-drs 14063
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