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

Theorem drsbn0 14322
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  e. 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 2388 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 14319 . 2  |-  ( K  e. Dirset 
<->  ( K  e.  Preset  /\  B  =/=  (/)  /\  A. x  e.  B  A. y  e.  B  E. z  e.  B  (
x ( le `  K ) z  /\  y ( le `  K ) z ) ) )
43simp2bi 973 1  |-  ( K  e. Dirset  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   (/)c0 3572   class class class wbr 4154   ` cfv 5395   Basecbs 13397   lecple 13464    Preset cpreset 14311  Dirsetcdrs 14312
This theorem is referenced by:  drsdirfi  14323  isipodrs  14515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-nul 4280
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-drs 14314
  Copyright terms: Public domain W3C validator