Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > drsbn0 | Structured version Visualization version GIF version |
Description: The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
drsbn0.b | ⊢ 𝐵 = (Base‘𝐾) |
Ref | Expression |
---|---|
drsbn0 | ⊢ (𝐾 ∈ Dirset → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drsbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | isdrs 17544 | . 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧))) |
4 | 3 | simp2bi 1142 | 1 ⊢ (𝐾 ∈ Dirset → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 Proset cproset 17536 Dirsetcdrs 17537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-drs 17539 |
This theorem is referenced by: drsdirfi 17548 isipodrs 17771 |
Copyright terms: Public domain | W3C validator |