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.

Step	Hyp	Ref	Expression
1		drsbn0.b	. . 3 |- B = ( Base ` K )
2		eqid 2451	. . 3 |- ( le ` K ) = ( le ` K )
3	1, 2	isdrs 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 ) ) )
4	3	simp2bi 1004	1 |- ( K e. Dirset -> B =/= (/) )

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