Theorem drsbn0 18272

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 2730	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3	1, 2	isdrs 18269	. 2 ⊢ (𝐾 ∈ Dirset ↔ (𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 (𝑥(le‘𝐾)𝑧 ∧ 𝑦(le‘𝐾)𝑧)))
4	3	simp2bi 1146	1 ⊢ (𝐾 ∈ Dirset → 𝐵 ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  ∅c0 4299   class class class wbr 5110  ‘cfv 6514  Basecbs 17186  lecple 17234   Proset cproset 18260  Dirsetcdrs 18261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-drs 18263

This theorem is referenced by:  drsdirfi  18273  isipodrs  18503