Theorem drsprs 18263

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 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2737	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18261	. 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 1146	1 ⊢ (𝐾 ∈ Dirset → 𝐾 ∈ Proset )

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  ∅c0 4274   class class class wbr 5086  ‘cfv 6493  Basecbs 17173  lecple 17221   Proset cproset 18252  Dirsetcdrs 18253

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-drs 18255

This theorem is referenced by:  drsdirfi  18265  isdrs2  18266