Theorem drsprs 18358

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 2769	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2769	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18356	. 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 1161	1 ⊢ (𝐾 ∈ Dirset → 𝐾 ∈ Proset )

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  ∅c0 4294   class class class wbr 5113  ‘cfv 6537  Basecbs 17268  lecple 17316   Proset cproset 18347  Dirsetcdrs 18348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-drs 18350

This theorem is referenced by:  drsdirfi  18360  isdrs2  18361