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Theorem drsbn0 17559
 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 2798 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2isdrs 17556 . 2 (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥(le‘𝐾)𝑧𝑦(le‘𝐾)𝑧)))
43simp2bi 1143 1 (𝐾 ∈ Dirset → 𝐵 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∅c0 4246   class class class wbr 5034  ‘cfv 6332  Basecbs 16495  lecple 16584   Proset cproset 17548  Dirsetcdrs 17549 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5178 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-drs 17551 This theorem is referenced by:  drsdirfi  17560  isipodrs  17783
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