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Theorem drsdir 15216
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.
StepHypRef Expression
1 isdrs.b . . . . 5  |-  B  =  ( Base `  K
)
2 isdrs.l . . . . 5  |-  .<_  =  ( le `  K )
31, 2isdrs 15215 . . . 4  |-  ( K  e. Dirset 
<->  ( K  e.  Preset  /\  B  =/=  (/)  /\  A. x  e.  B  A. y  e.  B  E. z  e.  B  (
x  .<_  z  /\  y  .<_  z ) ) )
43simp3bi 1005 . . 3  |-  ( K  e. Dirset  ->  A. x  e.  B  A. y  e.  B  E. z  e.  B  ( x  .<_  z  /\  y  .<_  z ) )
5 breq1 4396 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
65anbi1d 704 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  y  .<_  z ) ) )
76rexbidv 2855 . . . 4  |-  ( x  =  X  ->  ( E. z  e.  B  ( x  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  y  .<_  z ) ) )
8 breq1 4396 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
98anbi2d 703 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  Y  .<_  z ) ) )
109rexbidv 2855 . . . 4  |-  ( y  =  Y  ->  ( E. z  e.  B  ( X  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
117, 10rspc2v 3179 . . 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 ) ) )
124, 11syl5com 30 . 2  |-  ( K  e. Dirset  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
13123impib 1186 1  |-  ( ( K  e. Dirset  /\  X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   (/)c0 3738   class class class wbr 4393   ` cfv 5519   Basecbs 14285   lecple 14356    Preset cpreset 15207  Dirsetcdrs 15208
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 4522
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 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-drs 15210
This theorem is referenced by:  drsdirfi  15219
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