Theorem drsbn0 18504

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 2763	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18501	. 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 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∅c0 4281   class class class wbr 5106  ‘cfv 6555  Basecbs 17413  lecple 17461   Proset cproset 18492  Dirsetcdrs 18493

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-drs 18495

This theorem is used by:  drsdirfi  18505  isipodrs  18737