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Theorem isline 35876
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 35875 . . 3  |-  ( K  e.  D  ->  N  =  { x  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
65eleq2d 2452 . 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 5784 . . . . . . . . 9  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2466 . . . . . . . 8  |-  A  e. 
_V
98rabex 4516 . . . . . . 7  |-  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  e.  _V
10 eleq1 2454 . . . . . . 7  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  ( X  e.  _V  <->  { p  e.  A  |  p  .<_  ( q 
.\/  r ) }  e.  _V ) )
119, 10mpbiri 233 . . . . . 6  |-  ( X  =  { p  e.  A  |  p  .<_  ( q  .\/  r ) }  ->  X  e.  _V )
1211adantl 464 . . . . 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 2879 . . 3  |-  ( E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  ->  X  e.  _V )
15 eqeq1 2386 . . . . 5  |-  ( x  =  X  ->  (
x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  <->  X  =  { p  e.  A  |  p  .<_  ( q 
.\/  r ) } ) )
1615anbi2d 701 . . . 4  |-  ( x  =  X  ->  (
( q  =/=  r  /\  x  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } )  <-> 
( q  =/=  r  /\  X  =  {
p  e.  A  |  p  .<_  ( q  .\/  r ) } ) ) )
17162rexbidv 2900 . . 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 3178 . 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 261 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 184    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367    =/= wne 2577   E.wrex 2733   {crab 2736   _Vcvv 3034   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   lecple 14709   joincjn 15690   Atomscatm 35401   Linesclines 35631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-lines 35638
This theorem is referenced by:  islinei  35877  linepsubN  35889  isline2  35911
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