|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > drsprs | Structured version Visualization version GIF version | ||
| Description: A directed set is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| drsprs | ⊢ (𝐾 ∈ Dirset → 𝐾 ∈ Proset ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 2 | eqid 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 3 | 1, 2 | isdrs 18347 | . 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 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Proset cproset 18338 Dirsetcdrs 18339 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-drs 18341 | 
| This theorem is referenced by: drsdirfi 18351 isdrs2 18352 | 
| Copyright terms: Public domain | W3C validator |