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 preset. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
drsprs | ⊢ (𝐾 ∈ Dirset → 𝐾 ∈ Preset ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2622 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | isdrs 16934 | . 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Preset ∧ (Base‘𝐾) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∃𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧))) |
4 | 3 | simp1bi 1076 | 1 ⊢ (𝐾 ∈ Dirset → 𝐾 ∈ Preset ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 Preset cpreset 16926 Dirsetcdrs 16927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-drs 16929 |
This theorem is referenced by: drsdirfi 16938 isdrs2 16939 |
Copyright terms: Public domain | W3C validator |