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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 2736 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | 1, 2 | isdrs 18189 | . 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧))) |
4 | 3 | simp2bi 1146 | 1 ⊢ (𝐾 ∈ Dirset → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ∅c0 4282 class class class wbr 5105 ‘cfv 6496 Basecbs 17082 lecple 17139 Proset cproset 18181 Dirsetcdrs 18182 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5263 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-iota 6448 df-fv 6504 df-drs 18184 |
This theorem is referenced by: drsdirfi 18193 isipodrs 18425 |
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