Theorem drsdir 16935

Description: Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

Ref	Expression
isdrs.b	|- B = ( Base ` K )
isdrs.l	|- .<_ = ( le ` K )

Assertion

Ref	Expression
drsdir	|- ( ( K e. Dirset /\ X e. B /\ Y e. B ) -> E. z e. B ( X .<_ z /\ Y .<_ z ) )

Distinct variable groups:   z, K   z, B   z, .<_   z, X   z, Y

Proof of Theorem drsdir

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isdrs.b	. . . . 5 |- B = ( Base ` K )
2		isdrs.l	. . . . 5 |- .<_ = ( le ` K )
3	1, 2	isdrs 16934	. . . 4 |- ( K e. Dirset <-> ( K e. Preset /\ B =/= (/) /\ A. x e. B A. y e. B E. z e. B ( x .<_ z /\ y .<_ z ) ) )
4	3	simp3bi 1078	. . 3 |- ( K e. Dirset -> A. x e. B A. y e. B E. z e. B ( x .<_ z /\ y .<_ z ) )
5		breq1 4656	. . . . . 6 |- ( x = X -> ( x .<_ z <-> X .<_ z ) )
6	5	anbi1d 741	. . . . 5 |- ( x = X -> ( ( x .<_ z /\ y .<_ z ) <-> ( X .<_ z /\ y .<_ z ) ) )
7	6	rexbidv 3052	. . . 4 |- ( x = X -> ( E. z e. B ( x .<_ z /\ y .<_ z ) <-> E. z e. B ( X .<_ z /\ y .<_ z ) ) )
8		breq1 4656	. . . . . 6 |- ( y = Y -> ( y .<_ z <-> Y .<_ z ) )
9	8	anbi2d 740	. . . . 5 |- ( y = Y -> ( ( X .<_ z /\ y .<_ z ) <-> ( X .<_ z /\ Y .<_ z ) ) )
10	9	rexbidv 3052	. . . 4 |- ( y = Y -> ( E. z e. B ( X .<_ z /\ y .<_ z ) <-> E. z e. B ( X .<_ z /\ Y .<_ z ) ) )
11	7, 10	rspc2v 3322	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( A. x e. B A. y e. B E. z e. B ( x .<_ z /\ y .<_ z ) -> E. z e. B ( X .<_ z /\ Y .<_ z ) ) )
12	4, 11	syl5com 31	. 2 |- ( K e. Dirset -> ( ( X e. B /\ Y e. B ) -> E. z e. B ( X .<_ z /\ Y .<_ z ) ) )
13	12	3impib 1262	1 |- ( ( K e. Dirset /\ X e. B /\ Y e. B ) -> E. z e. B ( X .<_ z /\ Y .<_ z ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Preset cpreset 16926  Dirsetcdrs 16927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-drs 16929

This theorem is referenced by:  drsdirfi  16938