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Theorem isline 33348
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 33347 . . 3  |-  ( K  e.  D  ->  N  =  { x  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
65eleq2d 2524 . 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 5897 . . . . . . . . 9  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2535 . . . . . . . 8  |-  A  e. 
_V
98rabex 4567 . . . . . . 7  |-  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  e.  _V
10 eleq1 2527 . . . . . . 7  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  ( X  e.  _V  <->  { p  e.  A  |  p  .<_  ( q 
.\/  r ) }  e.  _V ) )
119, 10mpbiri 241 . . . . . 6  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  X  e.  _V )
1211adantl 472 . . . . 5  |-  ( ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  ->  X  e.  _V )
1312a1i 11 . . . 4  |-  ( ( q  e.  A  /\  r  e.  A )  ->  ( ( q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } )  ->  X  e.  _V ) )
1413rexlimivv 2895 . . 3  |-  ( E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  ->  X  e.  _V )
15 eqeq1 2465 . . . . 5  |-  ( x  =  X  ->  (
x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) )
1615anbi2d 715 . . . 4  |-  ( x  =  X  ->  (
( q  =/=  r  /\  x  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
17162rexbidv 2919 . . 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 3203 . 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 269 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   {cab 2447    =/= wne 2632   E.wrex 2749   {crab 2752   _Vcvv 3056   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   lecple 15245   joincjn 16237   Atomscatm 32873   Linesclines 33103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-lines 33110
This theorem is referenced by:  islinei  33349  linepsubN  33361  isline2  33383
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