Theorem drsbn0 18366

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 2762	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18363	. 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧)))
4	3	simp2bi 1163	1 ⊢ (𝐾 ∈ Dirset → 𝐵 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  ∅c0 4285   class class class wbr 5108  ‘cfv 6536  Basecbs 17275  lecple 17323   Proset cproset 18354  Dirsetcdrs 18355

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-drs 18357

This theorem is used by:  drsdirfi  18367  isipodrs  18599