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Theorem drsbn0 15206
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 2451 . . 3  |-  ( le
`  K )  =  ( le `  K
)
31, 2isdrs 15203 . 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 1004 1  |-  ( K  e. Dirset  ->  B  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642   A.wral 2793   E.wrex 2794   (/)c0 3732   class class class wbr 4387   ` cfv 5513   Basecbs 14273   lecple 14344    Preset cpreset 15195  Dirsetcdrs 15196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4516
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-iota 5476  df-fv 5521  df-drs 15198
This theorem is referenced by:  drsdirfi  15207  isipodrs  15430
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