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Theorem drsprs 18361
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 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2735 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2isdrs 18359 . 2 (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ (Base‘𝐾) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∃𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑧𝑦(le‘𝐾)𝑧)))
43simp1bi 1144 1 (𝐾 ∈ Dirset → 𝐾 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wne 2938  wral 3059  wrex 3068  c0 4339   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305   Proset cproset 18350  Dirsetcdrs 18351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-drs 18353
This theorem is referenced by:  drsdirfi  18363  isdrs2  18364
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