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Theorem drsbn0 18192
Description: The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypothesis
Ref Expression
drsbn0.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
drsbn0 (𝐾 ∈ Dirset → 𝐵 ≠ ∅)

Proof of Theorem drsbn0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 drsbn0.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2736 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2isdrs 18189 . 2 (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥(le‘𝐾)𝑧𝑦(le‘𝐾)𝑧)))
43simp2bi 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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