Theorem drsprs 18260

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 2732	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2732	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18258	. 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 396   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  ∅c0 4322   class class class wbr 5148  ‘cfv 6543  Basecbs 17148  lecple 17208   Proset cproset 18250  Dirsetcdrs 18251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-drs 18253

This theorem is referenced by:  drsdirfi  18262  isdrs2  18263