Theorem drsprs 18294

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 2728	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2728	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18292	. 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 1143	1 ⊢ (𝐾 ∈ Dirset → 𝐾 ∈ Proset )

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  ∅c0 4323   class class class wbr 5148  ‘cfv 6548  Basecbs 17179  lecple 17239   Proset cproset 18284  Dirsetcdrs 18285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-drs 18287

This theorem is referenced by:  drsdirfi  18296  isdrs2  18297