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Theorem drsdir 14392
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 14391 . . . 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 974 . . 3  |-  ( K  e. Dirset  ->  A. x  e.  B  A. y  e.  B  E. z  e.  B  ( x  .<_  z  /\  y  .<_  z ) )
5 breq1 4215 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
65anbi1d 686 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  y  .<_  z ) ) )
76rexbidv 2726 . . . 4  |-  ( x  =  X  ->  ( E. z  e.  B  ( x  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  y  .<_  z ) ) )
8 breq1 4215 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
98anbi2d 685 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  Y  .<_  z ) ) )
109rexbidv 2726 . . . 4  |-  ( y  =  Y  ->  ( E. z  e.  B  ( X  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
117, 10rspc2v 3058 . . 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 28 . 2  |-  ( K  e. Dirset  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
13123impib 1151 1  |-  ( ( K  e. Dirset  /\  X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   (/)c0 3628   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536    Preset cpreset 14383  Dirsetcdrs 14384
This theorem is referenced by:  drsdirfi  14395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-drs 14386
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