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Theorem isline 30538
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 30537 . . 3  |-  ( K  e.  D  ->  N  =  { x  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
65eleq2d 2505 . 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 5744 . . . . . . . . 9  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2508 . . . . . . . 8  |-  A  e. 
_V
98rabex 4356 . . . . . . 7  |-  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  e.  _V
10 eleq1 2498 . . . . . . 7  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  ( X  e.  _V  <->  { p  e.  A  |  p  .<_  ( q 
.\/  r ) }  e.  _V ) )
119, 10mpbiri 226 . . . . . 6  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  X  e.  _V )
1211adantl 454 . . . . 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 2837 . . 3  |-  ( E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  ->  X  e.  _V )
15 eqeq1 2444 . . . . 5  |-  ( x  =  X  ->  (
x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) )
1615anbi2d 686 . . . 4  |-  ( x  =  X  ->  (
( q  =/=  r  /\  x  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
17162rexbidv 2750 . . 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 3091 . 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 254 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   E.wrex 2708   {crab 2711   _Vcvv 2958   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30063   Linesclines 30293
This theorem is referenced by:  islinei  30539  linepsubN  30551  isline2  30573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-lines 30300
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