Theorem drsprs 16936

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
2		eqid 2622	. . 3 |- ( le ` K ) = ( le ` K )
3	1, 2	isdrs 16934	. 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 ) ) )
4	3	simp1bi 1076	1 |- ( K e. Dirset -> K e. Preset )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Preset cpreset 16926  Dirsetcdrs 16927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-drs 16929

This theorem is referenced by:  drsdirfi  16938  isdrs2  16939