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Theorem ellines 24775
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 2796 . 2  |-  ( A  e. LinesEE  ->  A  e.  _V )
2 ovex 5883 . . . . . . 7  |-  ( pLine q )  e.  _V
3 eleq1 2343 . . . . . . 7  |-  ( A  =  ( pLine q
)  ->  ( A  e.  _V  <->  ( pLine q
)  e.  _V )
)
42, 3mpbiri 224 . . . . . 6  |-  ( A  =  ( pLine q
)  ->  A  e.  _V )
54adantl 452 . . . . 5  |-  ( ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V )
65rexlimivw 2663 . . . 4  |-  ( E. q  e.  ( EE
`  n ) ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V )
76a1i 10 . . 3  |-  ( ( n  e.  NN  /\  p  e.  ( EE `  n ) )  -> 
( E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) )  ->  A  e.  _V ) )
87rexlimivv 2672 . 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 2343 . . 3  |-  ( x  =  A  ->  (
x  e. LinesEE  <->  A  e. LinesEE ) )
10 eqeq1 2289 . . . . . 6  |-  ( x  =  A  ->  (
x  =  ( pLine q )  <->  A  =  ( pLine q ) ) )
1110anbi2d 684 . . . . 5  |-  ( x  =  A  ->  (
( p  =/=  q  /\  x  =  (
pLine q ) )  <-> 
( p  =/=  q  /\  A  =  (
pLine q ) ) ) )
1211rexbidv 2564 . . . 4  |-  ( x  =  A  ->  ( E. q  e.  ( EE `  n ) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) ) ) )
13122rexbidv 2586 . . 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 24762 . . . . . 6  |- LinesEE  =  ran Line
15 df-line2 24760 . . . . . . 7  |- Line  =  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
1615rneqi 4905 . . . . . 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 5930 . . . . . 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 2307 . . . . 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 2347 . . . 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 2271 . . . . 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 2549 . . . . . . 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 1570 . . . . . 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 1818 . . . . . . 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 2581 . . . . . . . 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 2694 . . . . . . . . . 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 2549 . . . . . . . . . 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 242 . . . . . . . . 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 1570 . . . . . . . 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 240 . . . . . . 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 630 . . . . . . . . . 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 630 . . . . . . . . . . 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 736 . . . . . . . . . . . . . 14  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  n  e.  NN )
33 simplrr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  e.  ( EE `  n ) )
34 simpll 730 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  -> 
q  e.  ( EE
`  n ) )
35 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( q  e.  ( EE `  n )  /\  ( n  e.  NN  /\  p  e.  ( EE `  n
) ) )  /\  p  =/=  q )  ->  p  =/=  q )
3633, 34, 353jca 1132 . . . . . . . . . . . . . 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 518 . . . . . . . . . . . . 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 962 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
q  e.  ( EE
`  n ) )
39 simpl 443 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  n  e.  NN )
40 simpr1 961 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  e.  ( EE `  n ) )
4138, 39, 40jca32 521 . . . . . . . . . . . . . 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 963 . . . . . . . . . . . . . 14  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  ->  p  =/=  q )
4341, 42jca 518 . . . . . . . . . . . . 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 180 . . . . . . . . . . . 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 676 . . . . . . . . . . 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 242 . . . . . . . . . 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 242 . . . . . . . . 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 24767 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  { x  |  x  Colinear  <. p ,  q
>. } )
49 opex 4237 . . . . . . . . . . . . . 14  |-  <. p ,  q >.  e.  _V
50 dfec2 6663 . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
5349, 52brcnv 4864 . . . . . . . . . . . . . 14  |-  ( <.
p ,  q >. `' 
Colinear  x  <->  x  Colinear  <. p ,  q >. )
5453abbii 2395 . . . . . . . . . . . . 13  |-  { x  |  <. p ,  q
>. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. p ,  q >. }
5551, 54eqtri 2303 . . . . . . . . . . . 12  |-  [ <. p ,  q >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. p ,  q >. }
5648, 55syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  [ <. p ,  q >. ] `'  Colinear  )
5756eqeq2d 2294 . . . . . . . . . 10  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( x  =  ( pLine q )  <->  x  =  [ <. p ,  q
>. ] `'  Colinear  ) )
5857pm5.32i 618 . . . . . . . . 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 630 . . . . . . . . 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 263 . . . . . . . 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 1571 . . . . . . 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 269 . . . . . 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 240 . . . . 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 240 . . . 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 240 . . 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 2844 . 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 342 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 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544   _Vcvv 2788   <.cop 3643   class class class wbr 4023   `'ccnv 4688   ran crn 4690   ` cfv 5255  (class class class)co 5858   {coprab 5859   [cec 6658   NNcn 9746   EEcee 24516    Colinear ccolin 24660  Linecline2 24757  LinesEEclines2 24759
This theorem is referenced by:  linethru  24776  hilbert1.1  24777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-recs 6388  df-rdg 6423  df-ec 6662  df-nn 9747  df-colinear 24664  df-line2 24760  df-lines2 24762
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