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Theorem drsbn0 14386
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 2435 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 14383 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   (/)c0 3620   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528    Preset cpreset 14375  Dirsetcdrs 14376
This theorem is referenced by:  drsdirfi  14387  isipodrs  14579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-drs 14378
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