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Theorem drsdir 14162
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 14161 . . . 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 972 . . 3  |-  ( K  e. Dirset  ->  A. x  e.  B  A. y  e.  B  E. z  e.  B  ( x  .<_  z  /\  y  .<_  z ) )
5 breq1 4105 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
65anbi1d 685 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  y  .<_  z ) ) )
76rexbidv 2640 . . . 4  |-  ( x  =  X  ->  ( E. z  e.  B  ( x  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  y  .<_  z ) ) )
8 breq1 4105 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
98anbi2d 684 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  Y  .<_  z ) ) )
109rexbidv 2640 . . . 4  |-  ( y  =  Y  ->  ( E. z  e.  B  ( X  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
117, 10rspc2v 2966 . . 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 26 . 2  |-  ( K  e. Dirset  ->  ( ( X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
13123impib 1149 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620   (/)c0 3531   class class class wbr 4102   ` cfv 5334   Basecbs 13239   lecple 13306    Preset cpreset 14153  Dirsetcdrs 14154
This theorem is referenced by:  drsdirfi  14165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4228
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-drs 14156
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