Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isline Unicode version

Theorem isline 29928
Description: The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
Hypotheses
Ref Expression
isline.l  |-  .<_  =  ( le `  K )
isline.j  |-  .\/  =  ( join `  K )
isline.a  |-  A  =  ( Atoms `  K )
isline.n  |-  N  =  ( Lines `  K )
Assertion
Ref Expression
isline  |-  ( K  e.  D  ->  ( X  e.  N  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) ) )
Distinct variable groups:    q, p, r, A    K, p, q, r    X, q, r
Allowed substitution hints:    D( r, q, p)    .\/ ( r, q, p)    .<_ ( r, q, p)    N( r, q, p)    X( p)

Proof of Theorem isline
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isline.l . . . 4  |-  .<_  =  ( le `  K )
2 isline.j . . . 4  |-  .\/  =  ( join `  K )
3 isline.a . . . 4  |-  A  =  ( Atoms `  K )
4 isline.n . . . 4  |-  N  =  ( Lines `  K )
51, 2, 3, 4lineset 29927 . . 3  |-  ( K  e.  D  ->  N  =  { x  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
65eleq2d 2350 . 2  |-  ( K  e.  D  ->  ( X  e.  N  <->  X  e.  { x  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  x  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) } ) )
7 fvex 5539 . . . . . . . . 9  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2353 . . . . . . . 8  |-  A  e. 
_V
98rabex 4165 . . . . . . 7  |-  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  e.  _V
10 eleq1 2343 . . . . . . 7  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  ( X  e.  _V  <->  { p  e.  A  |  p  .<_  ( q 
.\/  r ) }  e.  _V ) )
119, 10mpbiri 224 . . . . . 6  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  X  e.  _V )
1211adantl 452 . . . . 5  |-  ( ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  ->  X  e.  _V )
1312a1i 10 . . . 4  |-  ( ( q  e.  A  /\  r  e.  A )  ->  ( ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } )  ->  X  e.  _V ) )
1413rexlimivv 2672 . . 3  |-  ( E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  ->  X  e.  _V )
15 eqeq1 2289 . . . . 5  |-  ( x  =  X  ->  (
x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) )
1615anbi2d 684 . . . 4  |-  ( x  =  X  ->  (
( q  =/=  r  /\  x  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
17162rexbidv 2586 . . 3  |-  ( x  =  X  ->  ( E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  x  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
1814, 17elab3 2921 . 2  |-  ( X  e.  { x  |  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  x  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) }  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) )
196, 18syl6bb 252 1  |-  ( K  e.  D  ->  ( X  e.  N  <->  E. q  e.  A  E. r  e.  A  ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   Linesclines 29683
This theorem is referenced by:  islinei  29929  linepsubN  29941  isline2  29963
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-lines 29690
  Copyright terms: Public domain W3C validator