Theorem drsbn0 18478

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.

Step	Hyp	Ref	Expression
1		drsbn0.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2760	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18475	. 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧)))
4	3	simp2bi 1164	1 ⊢ (𝐾 ∈ Dirset → 𝐵 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∅c0 4278   class class class wbr 5102  ‘cfv 6538  Basecbs 17387  lecple 17435   Proset cproset 18466  Dirsetcdrs 18467

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-drs 18469

This theorem is used by:  drsdirfi  18479  isipodrs  18711