Theorem drsbn0 15765

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.

Step	Hyp	Ref	Expression
1		drsbn0.b	. . 3 |- B = ( Base ` K )
2		eqid 2454	. . 3 |- ( le ` K ) = ( le ` K )
3	1, 2	isdrs 15762	. 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 ) ) )
4	3	simp2bi 1010	1 |- ( K e. Dirset -> B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   (/)c0 3783   class class class wbr 4439   ` cfv 5570   Basecbs 14716   lecple 14791   Preset cpreset 15754  Dirsetcdrs 15755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-drs 15757

This theorem is referenced by:  drsdirfi  15766  isipodrs  15990