MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  drsbn0 Structured version   Visualization version   GIF version

Theorem drsbn0 17145
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 2771 . . 3 (le‘𝐾) = (le‘𝐾)
31, 2isdrs 17142 . 2 (𝐾 ∈ Dirset ↔ (𝐾 ∈ Preset ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥(le‘𝐾)𝑧𝑦(le‘𝐾)𝑧)))
43simp2bi 1140 1 (𝐾 ∈ Dirset → 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  c0 4063   class class class wbr 4786  cfv 6031  Basecbs 16064  lecple 16156   Preset cpreset 17134  Dirsetcdrs 17135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-drs 17137
This theorem is referenced by:  drsdirfi  17146  isipodrs  17369
  Copyright terms: Public domain W3C validator