MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  drsdir Unicode version

Theorem drsdir 14069
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 14068 . . . 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 4026 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  z  <->  X  .<_  z ) )
65anbi1d 685 . . . . 5  |-  ( x  =  X  ->  (
( x  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  y  .<_  z ) ) )
76rexbidv 2564 . . . 4  |-  ( x  =  X  ->  ( E. z  e.  B  ( x  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  y  .<_  z ) ) )
8 breq1 4026 . . . . . 6  |-  ( y  =  Y  ->  (
y  .<_  z  <->  Y  .<_  z ) )
98anbi2d 684 . . . . 5  |-  ( y  =  Y  ->  (
( X  .<_  z  /\  y  .<_  z )  <->  ( X  .<_  z  /\  Y  .<_  z ) ) )
109rexbidv 2564 . . . 4  |-  ( y  =  Y  ->  ( E. z  e.  B  ( X  .<_  z  /\  y  .<_  z )  <->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) ) )
117, 10rspc2v 2890 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215    Preset cpreset 14060  Dirsetcdrs 14061
This theorem is referenced by:  drsdirfi  14072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-drs 14063
  Copyright terms: Public domain W3C validator