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Theorem ellines 30969
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 3066 . 2  |-  ( A  e. LinesEE  ->  A  e.  _V )
2 ovex 6348 . . . . . . 7  |-  ( pLine q )  e.  _V
3 eleq1 2528 . . . . . . 7  |-  ( A  =  ( pLine q
)  ->  ( A  e.  _V  <->  ( pLine q
)  e.  _V )
)
42, 3mpbiri 241 . . . . . 6  |-  ( A  =  ( pLine q
)  ->  A  e.  _V )
54adantl 472 . . . . 5  |-  ( ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V )
65rexlimivw 2888 . . . 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 2896 . 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 2528 . . 3  |-  ( x  =  A  ->  (
x  e. LinesEE  <->  A  e. LinesEE ) )
10 eqeq1 2466 . . . . . 6  |-  ( x  =  A  ->  (
x  =  ( pLine q )  <->  A  =  ( pLine q ) ) )
1110anbi2d 715 . . . . 5  |-  ( x  =  A  ->  (
( p  =/=  q  /\  x  =  (
pLine q ) )  <-> 
( p  =/=  q  /\  A  =  (
pLine q ) ) ) )
1211rexbidv 2913 . . . 4  |-  ( x  =  A  ->  ( E. q  e.  ( EE `  n ) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) ) ) )
13122rexbidv 2920 . . 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 30956 . . . . . 6  |- LinesEE  =  ran Line
15 df-line2 30954 . . . . . . 7  |- Line  =  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
1615rneqi 5083 . . . . . 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 6411 . . . . . 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 2488 . . . . 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 2532 . . . 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 2450 . . . . 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 2755 . . . . . . 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 1730 . . . . . 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 1942 . . . . . . 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 2925 . . . . . . . 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 2957 . . . . . . . . . 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 2755 . . . . . . . . . 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 259 . . . . . . . . 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 1730 . . . . . . . 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 257 . . . . . . 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 659 . . . . . . . . . 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 659 . . . . . . . . . . 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 775 . . . . . . . . . . . . . 14  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  n  e.  NN )
33 simplrr 776 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  e.  ( EE `  n ) )
34 simpll 765 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  -> 
q  e.  ( EE
`  n ) )
35 simpr 467 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  =/=  q )
3633, 34, 353jca 1194 . . . . . . . . . . . . . 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 539 . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
q  e.  ( EE
`  n ) )
39 simpl 463 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  n  e.  NN )
40 simpr1 1020 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  e.  ( EE `  n ) )
4138, 39, 40jca32 542 . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  =/=  q )
4341, 42jca 539 . . . . . . . . . . . . 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 192 . . . . . . . . . . . 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 706 . . . . . . . . . . 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 259 . . . . . . . . . 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 259 . . . . . . . . 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 30961 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  { x  |  x  Colinear  <. p ,  q
>. } )
49 opex 4681 . . . . . . . . . . . . . 14  |-  <. p ,  q >.  e.  _V
50 dfec2 7397 . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
5349, 52brcnv 5039 . . . . . . . . . . . . . 14  |-  ( <.
p ,  q >. `' 
Colinear  x  <->  x  Colinear  <. p ,  q >. )
5453abbii 2578 . . . . . . . . . . . . 13  |-  { x  |  <. p ,  q
>. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. p ,  q >. }
5551, 54eqtri 2484 . . . . . . . . . . . 12  |-  [ <. p ,  q >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. p ,  q >. }
5648, 55syl6eqr 2514 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  [ <. p ,  q >. ] `'  Colinear  )
5756eqeq2d 2472 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( x  =  ( pLine q )  <->  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )
5857pm5.32i 647 . . . . . . . . 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 659 . . . . . . . . 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 280 . . . . . . . 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 1731 . . . . . . 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 286 . . . . . 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 257 . . . . 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 257 . . . 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 257 . . 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 3120 . 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 359 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 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   {cab 2448    =/= wne 2633   E.wrex 2750   _Vcvv 3057   <.cop 3986   class class class wbr 4418   `'ccnv 4855   ran crn 4857   ` cfv 5605  (class class class)co 6320   {coprab 6321   [cec 7392   NNcn 10642   EEcee 24974    Colinear ccolin 30854  Linecline2 30951  LinesEEclines2 30953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-i2m1 9638  ax-1ne0 9639  ax-rrecex 9642  ax-cnre 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-oprab 6324  df-om 6725  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-ec 7396  df-nn 10643  df-colinear 30856  df-line2 30954  df-lines2 30956
This theorem is referenced by:  linethru  30970  hilbert1.1  30971
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