Theorem drsprs 18264

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.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2729	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18262	. 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ (Base‘𝐾) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∃𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧)))
4	3	simp1bi 1145	1 ⊢ (𝐾 ∈ Dirset → 𝐾 ∈ Proset )

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∅c0 4296   class class class wbr 5107  ‘cfv 6511  Basecbs 17179  lecple 17227   Proset cproset 18253  Dirsetcdrs 18254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-drs 18256

This theorem is referenced by:  drsdirfi  18266  isdrs2  18267