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Theorem ellines 26086
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 2964 . 2  |-  ( A  e. LinesEE  ->  A  e.  _V )
2 ovex 6106 . . . . . . 7  |-  ( pLine q )  e.  _V
3 eleq1 2496 . . . . . . 7  |-  ( A  =  ( pLine q
)  ->  ( A  e.  _V  <->  ( pLine q
)  e.  _V )
)
42, 3mpbiri 225 . . . . . 6  |-  ( A  =  ( pLine q
)  ->  A  e.  _V )
54adantl 453 . . . . 5  |-  ( ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V )
65rexlimivw 2826 . . . 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 2835 . 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 2496 . . 3  |-  ( x  =  A  ->  (
x  e. LinesEE  <->  A  e. LinesEE ) )
10 eqeq1 2442 . . . . . 6  |-  ( x  =  A  ->  (
x  =  ( pLine q )  <->  A  =  ( pLine q ) ) )
1110anbi2d 685 . . . . 5  |-  ( x  =  A  ->  (
( p  =/=  q  /\  x  =  (
pLine q ) )  <-> 
( p  =/=  q  /\  A  =  (
pLine q ) ) ) )
1211rexbidv 2726 . . . 4  |-  ( x  =  A  ->  ( E. q  e.  ( EE `  n ) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) ) ) )
13122rexbidv 2748 . . 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 26073 . . . . . 6  |- LinesEE  =  ran Line
15 df-line2 26071 . . . . . . 7  |- Line  =  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
1615rneqi 5096 . . . . . 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 6156 . . . . . 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 2460 . . . . 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 2500 . . . 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 2424 . . . . 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 2711 . . . . . . 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 1593 . . . . . 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 1759 . . . . . . 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 2743 . . . . . . . 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 2862 . . . . . . . . . 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 2711 . . . . . . . . . 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 243 . . . . . . . . 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 1593 . . . . . . . 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 241 . . . . . . 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 631 . . . . . . . . . 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 631 . . . . . . . . . . 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 737 . . . . . . . . . . . . . 14  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  n  e.  NN )
33 simplrr 738 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  e.  ( EE `  n ) )
34 simpll 731 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  -> 
q  e.  ( EE
`  n ) )
35 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  =/=  q )
3633, 34, 353jca 1134 . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . 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 964 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
q  e.  ( EE
`  n ) )
39 simpl 444 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  n  e.  NN )
40 simpr1 963 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  e.  ( EE `  n ) )
4138, 39, 40jca32 522 . . . . . . . . . . . . . 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 965 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  =/=  q )
4341, 42jca 519 . . . . . . . . . . . . 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 181 . . . . . . . . . . . 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 677 . . . . . . . . . . 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 243 . . . . . . . . . 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 243 . . . . . . . . 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 26078 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  { x  |  x  Colinear  <. p ,  q
>. } )
49 opex 4427 . . . . . . . . . . . . . 14  |-  <. p ,  q >.  e.  _V
50 dfec2 6908 . . . . . . . . . . . . . 14  |-  ( <.
p ,  q >.  e.  _V  ->  [ <. p ,  q >. ] `'  Colinear  =  { x  |  <. p ,  q >. `'  Colinear  x } )
5149, 50ax-mp 8 . . . . . . . . . . . . 13  |-  [ <. p ,  q >. ] `'  Colinear  =  { x  |  <. p ,  q >. `'  Colinear  x }
52 vex 2959 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
5349, 52brcnv 5055 . . . . . . . . . . . . . 14  |-  ( <.
p ,  q >. `' 
Colinear  x  <->  x  Colinear  <. p ,  q >. )
5453abbii 2548 . . . . . . . . . . . . 13  |-  { x  |  <. p ,  q
>. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. p ,  q >. }
5551, 54eqtri 2456 . . . . . . . . . . . 12  |-  [ <. p ,  q >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. p ,  q >. }
5648, 55syl6eqr 2486 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  [ <. p ,  q >. ] `'  Colinear  )
5756eqeq2d 2447 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( x  =  ( pLine q )  <->  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )
5857pm5.32i 619 . . . . . . . . 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 631 . . . . . . . . 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 264 . . . . . . . 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 1594 . . . . . . 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 270 . . . . . 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 241 . . . . 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 241 . . . 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 241 . . 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 3012 . 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 343 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   E.wrex 2706   _Vcvv 2956   <.cop 3817   class class class wbr 4212   `'ccnv 4877   ran crn 4879   ` cfv 5454  (class class class)co 6081   {coprab 6082   [cec 6903   NNcn 10000   EEcee 25827    Colinear ccolin 25971  Linecline2 26068  LinesEEclines2 26070
This theorem is referenced by:  linethru  26087  hilbert1.1  26088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-i2m1 9058  ax-1ne0 9059  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-recs 6633  df-rdg 6668  df-ec 6907  df-nn 10001  df-colinear 25975  df-line2 26071  df-lines2 26073
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