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Theorem ellines 28188
Description: Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
ellines  |-  ( A  e. LinesEE 
<->  E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  A  =  ( pLine q ) ) )
Distinct variable group:    A, n, p, q

Proof of Theorem ellines
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2986 . 2  |-  ( A  e. LinesEE  ->  A  e.  _V )
2 ovex 6121 . . . . . . 7  |-  ( pLine q )  e.  _V
3 eleq1 2503 . . . . . . 7  |-  ( A  =  ( pLine q
)  ->  ( A  e.  _V  <->  ( pLine q
)  e.  _V )
)
42, 3mpbiri 233 . . . . . 6  |-  ( A  =  ( pLine q
)  ->  A  e.  _V )
54adantl 466 . . . . 5  |-  ( ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V )
65rexlimivw 2842 . . . 4  |-  ( E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V )
76a1i 11 . . 3  |-  ( ( n  e.  NN  /\  p  e.  ( EE `  n ) )  -> 
( E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V ) )
87rexlimivv 2851 . 2  |-  ( E. n  e.  NN  E. p  e.  ( EE `  n ) E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V )
9 eleq1 2503 . . 3  |-  ( x  =  A  ->  (
x  e. LinesEE  <->  A  e. LinesEE ) )
10 eqeq1 2449 . . . . . 6  |-  ( x  =  A  ->  (
x  =  ( pLine q )  <->  A  =  ( pLine q ) ) )
1110anbi2d 703 . . . . 5  |-  ( x  =  A  ->  (
( p  =/=  q  /\  x  =  (
pLine q ) )  <-> 
( p  =/=  q  /\  A  =  (
pLine q ) ) ) )
1211rexbidv 2741 . . . 4  |-  ( x  =  A  ->  ( E. q  e.  ( EE `  n ) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) ) ) )
13122rexbidv 2763 . . 3  |-  ( x  =  A  ->  ( E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. n  e.  NN  E. p  e.  ( EE `  n
) E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) ) ) )
14 df-lines2 28175 . . . . . 6  |- LinesEE  =  ran Line
15 df-line2 28173 . . . . . . 7  |- Line  =  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
1615rneqi 5071 . . . . . 6  |-  ran Line  =  ran  {
<. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
17 rnoprab 6178 . . . . . 6  |-  ran  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }  =  { x  |  E. p E. q E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
1814, 16, 173eqtri 2467 . . . . 5  |- LinesEE  =  {
x  |  E. p E. q E. n  e.  NN  ( ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n
)  /\  p  =/=  q )  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
1918eleq2i 2507 . . . 4  |-  ( x  e. LinesEE 
<->  x  e.  { x  |  E. p E. q E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) } )
20 abid 2431 . . . . 5  |-  ( x  e.  { x  |  E. p E. q E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }  <->  E. p E. q E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )
21 df-rex 2726 . . . . . . 7  |-  ( E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  )  <->  E. n
( n  e.  NN  /\  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) ) )
22212exbii 1635 . . . . . 6  |-  ( E. p E. q E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  )  <->  E. p E. q E. n ( n  e.  NN  /\  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) ) )
23 exrot3 1790 . . . . . . 7  |-  ( E. n E. p E. q ( q  e.  ( EE `  n
)  /\  ( (
n  e.  NN  /\  p  e.  ( EE `  n ) )  /\  ( p  =/=  q  /\  x  =  (
pLine q ) ) ) )  <->  E. p E. q E. n ( q  e.  ( EE
`  n )  /\  ( ( n  e.  NN  /\  p  e.  ( EE `  n
) )  /\  (
p  =/=  q  /\  x  =  ( pLine q ) ) ) ) )
24 r2ex 2758 . . . . . . . 8  |-  ( E. n  e.  NN  E. p  e.  ( EE `  n ) E. q  e.  ( EE `  n
) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. n E. p
( ( n  e.  NN  /\  p  e.  ( EE `  n
) )  /\  E. q  e.  ( EE `  n ) ( p  =/=  q  /\  x  =  ( pLine q
) ) ) )
25 r19.42v 2880 . . . . . . . . . 10  |-  ( E. q  e.  ( EE
`  n ) ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  ( p  =/=  q  /\  x  =  ( pLine q ) ) )  <->  ( (
n  e.  NN  /\  p  e.  ( EE `  n ) )  /\  E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  x  =  ( pLine q ) ) ) )
26 df-rex 2726 . . . . . . . . . 10  |-  ( E. q  e.  ( EE
`  n ) ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  ( p  =/=  q  /\  x  =  ( pLine q ) ) )  <->  E. q
( q  e.  ( EE `  n )  /\  ( ( n  e.  NN  /\  p  e.  ( EE `  n
) )  /\  (
p  =/=  q  /\  x  =  ( pLine q ) ) ) ) )
2725, 26bitr3i 251 . . . . . . . . 9  |-  ( ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  E. q  e.  ( EE `  n
) ( p  =/=  q  /\  x  =  ( pLine q ) ) )  <->  E. q
( q  e.  ( EE `  n )  /\  ( ( n  e.  NN  /\  p  e.  ( EE `  n
) )  /\  (
p  =/=  q  /\  x  =  ( pLine q ) ) ) ) )
28272exbii 1635 . . . . . . . 8  |-  ( E. n E. p ( ( n  e.  NN  /\  p  e.  ( EE
`  n ) )  /\  E. q  e.  ( EE `  n
) ( p  =/=  q  /\  x  =  ( pLine q ) ) )  <->  E. n E. p E. q ( q  e.  ( EE
`  n )  /\  ( ( n  e.  NN  /\  p  e.  ( EE `  n
) )  /\  (
p  =/=  q  /\  x  =  ( pLine q ) ) ) ) )
2924, 28bitri 249 . . . . . . 7  |-  ( E. n  e.  NN  E. p  e.  ( EE `  n ) E. q  e.  ( EE `  n
) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. n E. p E. q ( q  e.  ( EE `  n
)  /\  ( (
n  e.  NN  /\  p  e.  ( EE `  n ) )  /\  ( p  =/=  q  /\  x  =  (
pLine q ) ) ) ) )
30 anass 649 . . . . . . . . . 10  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  ( p  =/=  q  /\  x  =  (
pLine q ) ) )  <->  ( q  e.  ( EE `  n
)  /\  ( (
n  e.  NN  /\  p  e.  ( EE `  n ) )  /\  ( p  =/=  q  /\  x  =  (
pLine q ) ) ) ) )
31 anass 649 . . . . . . . . . . 11  |-  ( ( ( ( q  e.  ( EE `  n
)  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  /\  x  =  ( pLine q ) )  <->  ( (
q  e.  ( EE
`  n )  /\  ( n  e.  NN  /\  p  e.  ( EE
`  n ) ) )  /\  ( p  =/=  q  /\  x  =  ( pLine q
) ) ) )
32 simplrl 759 . . . . . . . . . . . . . 14  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  n  e.  NN )
33 simplrr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  e.  ( EE `  n ) )
34 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  -> 
q  e.  ( EE
`  n ) )
35 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  =/=  q )
3633, 34, 353jca 1168 . . . . . . . . . . . . . 14  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  -> 
( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
) )
3732, 36jca 532 . . . . . . . . . . . . 13  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  -> 
( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
) ) )
38 simpr2 995 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
q  e.  ( EE
`  n ) )
39 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  n  e.  NN )
40 simpr1 994 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  e.  ( EE `  n ) )
4138, 39, 40jca32 535 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) ) )
42 simpr3 996 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  =/=  q )
4341, 42jca 532 . . . . . . . . . . . . 13  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( ( q  e.  ( EE `  n
)  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q ) )
4437, 43impbii 188 . . . . . . . . . . . 12  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  <->  ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n
)  /\  p  =/=  q ) ) )
4544anbi1i 695 . . . . . . . . . . 11  |-  ( ( ( ( q  e.  ( EE `  n
)  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  /\  x  =  ( pLine q ) )  <->  ( (
n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  /\  x  =  ( pLine q ) ) )
4631, 45bitr3i 251 . . . . . . . . . 10  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  ( p  =/=  q  /\  x  =  (
pLine q ) ) )  <->  ( ( n  e.  NN  /\  (
p  e.  ( EE
`  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  /\  x  =  ( pLine q ) ) )
4730, 46bitr3i 251 . . . . . . . . 9  |-  ( ( q  e.  ( EE
`  n )  /\  ( ( n  e.  NN  /\  p  e.  ( EE `  n
) )  /\  (
p  =/=  q  /\  x  =  ( pLine q ) ) ) )  <->  ( ( n  e.  NN  /\  (
p  e.  ( EE
`  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  /\  x  =  ( pLine q ) ) )
48 fvline 28180 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  { x  |  x  Colinear  <. p ,  q
>. } )
49 opex 4561 . . . . . . . . . . . . . 14  |-  <. p ,  q >.  e.  _V
50 dfec2 7109 . . . . . . . . . . . . . 14  |-  ( <.
p ,  q >.  e.  _V  ->  [ <. p ,  q >. ] `'  Colinear  =  { x  |  <. p ,  q >. `'  Colinear  x } )
5149, 50ax-mp 5 . . . . . . . . . . . . 13  |-  [ <. p ,  q >. ] `'  Colinear  =  { x  |  <. p ,  q >. `'  Colinear  x }
52 vex 2980 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
5349, 52brcnv 5027 . . . . . . . . . . . . . 14  |-  ( <.
p ,  q >. `' 
Colinear  x  <->  x  Colinear  <. p ,  q >. )
5453abbii 2560 . . . . . . . . . . . . 13  |-  { x  |  <. p ,  q
>. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. p ,  q >. }
5551, 54eqtri 2463 . . . . . . . . . . . 12  |-  [ <. p ,  q >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. p ,  q >. }
5648, 55syl6eqr 2493 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  [ <. p ,  q >. ] `'  Colinear  )
5756eqeq2d 2454 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( x  =  ( pLine q )  <->  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )
5857pm5.32i 637 . . . . . . . . 9  |-  ( ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
) )  /\  x  =  ( pLine q
) )  <->  ( (
n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  /\  x  =  [ <. p ,  q >. ] `'  Colinear  ) )
59 anass 649 . . . . . . . . 9  |-  ( ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
) )  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  )  <->  ( n  e.  NN  /\  ( ( p  e.  ( EE
`  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q )  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) ) )
6047, 58, 593bitrri 272 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )  <-> 
( q  e.  ( EE `  n )  /\  ( ( n  e.  NN  /\  p  e.  ( EE `  n
) )  /\  (
p  =/=  q  /\  x  =  ( pLine q ) ) ) ) )
61603exbii 1636 . . . . . . 7  |-  ( E. p E. q E. n ( n  e.  NN  /\  ( ( p  e.  ( EE
`  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q )  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )  <->  E. p E. q E. n ( q  e.  ( EE `  n
)  /\  ( (
n  e.  NN  /\  p  e.  ( EE `  n ) )  /\  ( p  =/=  q  /\  x  =  (
pLine q ) ) ) ) )
6223, 29, 613bitr4ri 278 . . . . . 6  |-  ( E. p E. q E. n ( n  e.  NN  /\  ( ( p  e.  ( EE
`  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q )  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )  <->  E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  x  =  ( pLine q ) ) )
6322, 62bitri 249 . . . . 5  |-  ( E. p E. q E. n  e.  NN  (
( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  )  <->  E. n  e.  NN  E. p  e.  ( EE `  n
) E. q  e.  ( EE `  n
) ( p  =/=  q  /\  x  =  ( pLine q ) ) )
6420, 63bitri 249 . . . 4  |-  ( x  e.  { x  |  E. p E. q E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }  <->  E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  x  =  ( pLine q ) ) )
6519, 64bitri 249 . . 3  |-  ( x  e. LinesEE 
<->  E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  x  =  ( pLine q ) ) )
669, 13, 65vtoclbg 3036 . 2  |-  ( A  e.  _V  ->  ( A  e. LinesEE  <->  E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  A  =  ( pLine q ) ) ) )
671, 8, 66pm5.21nii 353 1  |-  ( A  e. LinesEE 
<->  E. n  e.  NN  E. p  e.  ( EE
`  n ) E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  A  =  ( pLine q ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2611   E.wrex 2721   _Vcvv 2977   <.cop 3888   class class class wbr 4297   `'ccnv 4844   ran crn 4846   ` cfv 5423  (class class class)co 6096   {coprab 6097   [cec 7104   NNcn 10327   EEcee 23139    Colinear ccolin 28073  Linecline2 28170  LinesEEclines2 28172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-i2m1 9355  ax-1ne0 9356  ax-rrecex 9359  ax-cnre 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-om 6482  df-recs 6837  df-rdg 6871  df-ec 7108  df-nn 10328  df-colinear 28075  df-line2 28173  df-lines2 28175
This theorem is referenced by:  linethru  28189  hilbert1.1  28190
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