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Theorem drsprs 17548
Description: A directed set is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
drsprs (𝐾 ∈ Dirset → 𝐾 ∈ Proset )

Proof of Theorem drsprs
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2823 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2823 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2isdrs 17546 . 2 (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ (Base‘𝐾) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∃𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑧𝑦(le‘𝐾)𝑧)))
43simp1bi 1141 1 (𝐾 ∈ Dirset → 𝐾 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2114  wne 3018  wral 3140  wrex 3141  c0 4293   class class class wbr 5068  cfv 6357  Basecbs 16485  lecple 16574   Proset cproset 17538  Dirsetcdrs 17539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-drs 17541
This theorem is referenced by:  drsdirfi  17550  isdrs2  17551
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