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Theorem ellines 32259
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 3212 . 2 |- ( A e. LinesEE -> A e. _V )
2 ovex 6678 . . . . . . 7 |- ( pLine q ) e. _V
3 eleq1 2689 . . . . . . 7 |- ( A = ( pLine q ) -> ( A e. _V <-> ( pLine q ) e. _V ) )
42, 3mpbiri 248 . . . . . 6 |- ( A = ( pLine q ) -> A e. _V )
54adantl 482 . . . . 5 |- ( ( p =/= q /\ A = ( pLine q ) ) -> A e. _V )
65rexlimivw 3029 . . . 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 3036 . 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 2689 . . 3 |- ( x = A -> ( x e. LinesEE <-> A e. LinesEE ) )
10 eqeq1 2626 . . . . . 6 |- ( x = A -> ( x = ( pLine q ) <-> A = ( pLine q ) ) )
1110anbi2d 740 . . . . 5 |- ( x = A -> ( ( p =/= q /\ x = ( pLine q ) ) <-> ( p =/= q /\ A = ( pLine q ) ) ) )
1211rexbidv 3052 . . . 4 |- ( x = A -> ( E. q e. ( EE ` n ) ( p =/= q /\ x = ( pLine q ) ) <-> E. q e. ( EE ` n ) ( p =/= q /\ A = ( pLine q ) ) ) )
13122rexbidv 3057 . . 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 32246 . . . . . 6 |- LinesEE = ran Line
15 df-line2 32244 . . . . . . 7 |- Line = { <. <. p , q >. , x >. | E. n e. NN ( ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) /\ x = [ <. p , q >. ] `' Colinear ) }
1615rneqi 5352 . . . . . 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 6743 . . . . . 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 2648 . . . . 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 2693 . . . 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 2610 . . . . 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 2918 . . . . . . 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 1775 . . . . . 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 2045 . . . . . . 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 3061 . . . . . . . 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 3092 . . . . . . . . . 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 2918 . . . . . . . . . 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 266 . . . . . . . . 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 1775 . . . . . . . 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 264 . . . . . . 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 681 . . . . . . . . . 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 681 . . . . . . . . . . 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 800 . . . . . . . . . . . . . 14 |- ( ( ( q e. ( EE ` n ) /\ ( n e. NN /\ p e. ( EE ` n ) ) ) /\ p =/= q ) -> n e. NN )
33 simplrr 801 . . . . . . . . . . . . . . 15 |- ( ( ( q e. ( EE ` n ) /\ ( n e. NN /\ p e. ( EE ` n ) ) ) /\ p =/= q ) -> p e. ( EE ` n ) )
34 simpll 790 . . . . . . . . . . . . . . 15 |- ( ( ( q e. ( EE ` n ) /\ ( n e. NN /\ p e. ( EE ` n ) ) ) /\ p =/= q ) -> q e. ( EE ` n ) )
35 simpr 477 . . . . . . . . . . . . . . 15 |- ( ( ( q e. ( EE ` n ) /\ ( n e. NN /\ p e. ( EE ` n ) ) ) /\ p =/= q ) -> p =/= q )
3633, 34, 353jca 1242 . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . . 15 |- ( ( n e. NN /\ ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) ) -> q e. ( EE ` n ) )
39 simpl 473 . . . . . . . . . . . . . . 15 |- ( ( n e. NN /\ ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) ) -> n e. NN )
40 simpr1 1067 . . . . . . . . . . . . . . 15 |- ( ( n e. NN /\ ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) ) -> p e. ( EE ` n ) )
4138, 39, 40jca32 558 . . . . . . . . . . . . . 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 1069 . . . . . . . . . . . . . 14 |- ( ( n e. NN /\ ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) ) -> p =/= q )
4341, 42jca 554 . . . . . . . . . . . . 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 199 . . . . . . . . . . . 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 731 . . . . . . . . . . 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 266 . . . . . . . . . 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 266 . . . . . . . . 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 32251 . . . . . . . . . . . 12 |- ( ( n e. NN /\ ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) ) -> ( pLine q ) = { x | x Colinear <. p , q >. } )
49 opex 4932 . . . . . . . . . . . . . 14 |- <. p , q >. e. _V
50 dfec2 7745 . . . . . . . . . . . . . 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 3203 . . . . . . . . . . . . . . 15 |- x e. _V
5349, 52brcnv 5305 . . . . . . . . . . . . . 14 |- ( <. p , q >. `' Colinear x <-> x Colinear <. p , q >. )
5453abbii 2739 . . . . . . . . . . . . 13 |- { x | <. p , q >. `' Colinear x } = { x | x Colinear <. p , q >. }
5551, 54eqtri 2644 . . . . . . . . . . . 12 |- [ <. p , q >. ] `' Colinear = { x | x Colinear <. p , q >. }
5648, 55syl6eqr 2674 . . . . . . . . . . 11 |- ( ( n e. NN /\ ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) ) -> ( pLine q ) = [ <. p , q >. ] `' Colinear )
5756eqeq2d 2632 . . . . . . . . . 10 |- ( ( n e. NN /\ ( p e. ( EE ` n ) /\ q e. ( EE ` n ) /\ p =/= q ) ) -> ( x = ( pLine q ) <-> x = [ <. p , q >. ] `' Colinear ) )
5857pm5.32i 669 . . . . . . . . 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 681 . . . . . . . . 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 287 . . . . . . . 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 1776 . . . . . . 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 293 . . . . . 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 264 . . . . 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 264 . . . 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 264 . . 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 3267 . 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 368 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   _Vcvv 3200   <.cop 4183   class class class wbr 4653   `'ccnv 5113   ran crn 5115   ` cfv 5888  (class class class)co 6650   {coprab 6651   [cec 7740   NNcn 11020   EEcee 25768   Colinear ccolin 32144  Linecline2 32241  LinesEEclines2 32243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-ec 7744  df-nn 11021  df-colinear 32146  df-line2 32244  df-lines2 32246
This theorem is referenced by:  linethru  32260  hilbert1.1  32261
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