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Theorem ellines 25990
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 2924 . 2  |-  ( A  e. LinesEE  ->  A  e.  _V )
2 ovex 6065 . . . . . . 7  |-  ( pLine q )  e.  _V
3 eleq1 2464 . . . . . . 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 2786 . . . 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 2795 . 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 2464 . . 3  |-  ( x  =  A  ->  (
x  e. LinesEE  <->  A  e. LinesEE ) )
10 eqeq1 2410 . . . . . 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 2687 . . . 4  |-  ( x  =  A  ->  ( E. q  e.  ( EE `  n ) ( p  =/=  q  /\  x  =  ( pLine q ) )  <->  E. q  e.  ( EE `  n
) ( p  =/=  q  /\  A  =  ( pLine q ) ) ) )
13122rexbidv 2709 . . 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 25977 . . . . . 6  |- LinesEE  =  ran Line
15 df-line2 25975 . . . . . . 7  |- Line  =  { <. <. p ,  q
>. ,  x >.  |  E. n  e.  NN  ( ( p  e.  ( EE `  n
)  /\  q  e.  ( EE `  n )  /\  p  =/=  q
)  /\  x  =  [ <. p ,  q
>. ] `'  Colinear  ) }
1615rneqi 5055 . . . . . 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 6115 . . . . . 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 2428 . . . . 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 2468 . . . 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 2392 . . . . 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 2672 . . . . . . 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 1590 . . . . . 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 1755 . . . . . . 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 2704 . . . . . . . 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 2822 . . . . . . . . . 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 2672 . . . . . . . . . 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 1590 . . . . . . . 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 25982 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  { x  |  x  Colinear  <. p ,  q
>. } )
49 opex 4387 . . . . . . . . . . . . . 14  |-  <. p ,  q >.  e.  _V
50 dfec2 6867 . . . . . . . . . . . . . 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 2919 . . . . . . . . . . . . . . 15  |-  x  e. 
_V
5349, 52brcnv 5014 . . . . . . . . . . . . . 14  |-  ( <.
p ,  q >. `' 
Colinear  x  <->  x  Colinear  <. p ,  q >. )
5453abbii 2516 . . . . . . . . . . . . 13  |-  { x  |  <. p ,  q
>. `' 
Colinear  x }  =  {
x  |  x  Colinear  <. p ,  q >. }
5551, 54eqtri 2424 . . . . . . . . . . . 12  |-  [ <. p ,  q >. ] `'  Colinear  =  { x  |  x 
Colinear 
<. p ,  q >. }
5648, 55syl6eqr 2454 . . . . . . . . . . 11  |-  ( ( n  e.  NN  /\  ( p  e.  ( EE `  n )  /\  q  e.  ( EE `  n )  /\  p  =/=  q ) )  -> 
( pLine q )  =  [ <. p ,  q >. ] `'  Colinear  )
5756eqeq2d 2415 . . . . . . . . . 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 1591 . . . . . . 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 2972 . 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 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   E.wrex 2667   _Vcvv 2916   <.cop 3777   class class class wbr 4172   `'ccnv 4836   ran crn 4838   ` cfv 5413  (class class class)co 6040   {coprab 6041   [cec 6862   NNcn 9956   EEcee 25731    Colinear ccolin 25875  Linecline2 25972  LinesEEclines2 25974
This theorem is referenced by:  linethru  25991  hilbert1.1  25992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-i2m1 9014  ax-1ne0 9015  ax-rrecex 9018  ax-cnre 9019
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-recs 6592  df-rdg 6627  df-ec 6866  df-nn 9957  df-colinear 25879  df-line2 25975  df-lines2 25977
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