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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 2737 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | isdrs 17808 | . 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧))) |
4 | 3 | simp2bi 1148 | 1 ⊢ (𝐾 ∈ Dirset → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 ∅c0 4237 class class class wbr 5053 ‘cfv 6380 Basecbs 16760 lecple 16809 Proset cproset 17800 Dirsetcdrs 17801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-nul 5199 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-drs 17803 |
This theorem is referenced by: drsdirfi 17812 isipodrs 18043 |
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