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Theorem drsdir 15763
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 15762 . . . 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 1011 . . 3  |-  ( K  e. Dirset  ->  A. x  e.  B  A. y  e.  B  E. z  e.  B  ( x  .<_  z  /\  y  .<_  z ) )
5 breq1 4442 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
65anbi1d 702 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  y  .<_  z ) ) )
76rexbidv 2965 . . . 4  |-  ( x  =  X  ->  ( E. z  e.  B  ( x  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  y  .<_  z ) ) )
8 breq1 4442 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
98anbi2d 701 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  Y  .<_  z ) ) )
109rexbidv 2965 . . . 4  |-  ( y  =  Y  ->  ( E. z  e.  B  ( X  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
117, 10rspc2v 3216 . . 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 1192 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 367    /\ w3a 971    = 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
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