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Theorem isline 29746
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 29745 . . 3  |-  ( K  e.  D  ->  N  =  { x  |  E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  x  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } ) } )
65eleq2d 2383 . 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 5577 . . . . . . . . 9  |-  ( Atoms `  K )  e.  _V
83, 7eqeltri 2386 . . . . . . . 8  |-  A  e. 
_V
98rabex 4202 . . . . . . 7  |-  { p  e.  A  |  p  .<_  ( q  .\/  r
) }  e.  _V
10 eleq1 2376 . . . . . . 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 2706 . . 3  |-  ( E. q  e.  A  E. r  e.  A  (
q  =/=  r  /\  X  =  { p  e.  A  |  p  .<_  ( q  .\/  r
) } )  ->  X  e.  _V )
15 eqeq1 2322 . . . . 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 2620 . . 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 2955 . 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 1633    e. wcel 1701   {cab 2302    =/= wne 2479   E.wrex 2578   {crab 2581   _Vcvv 2822   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   lecple 13262   joincjn 14127   Atomscatm 29271   Linesclines 29501
This theorem is referenced by:  islinei  29747  linepsubN  29759  isline2  29781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-lines 29508
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