Theorem linedegen 32250

Description: When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
linedegen	|- ( ALine A ) = (/)

Proof of Theorem linedegen

Dummy variables l n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( ALine A ) = (Line ` <. A , A >. )
2		neirr 2803	. . . . . . . . . . 11 |- -. A =/= A
3		simp3 1063	. . . . . . . . . . 11 |- ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) -> A =/= A )
4	2, 3	mto 188	. . . . . . . . . 10 |- -. ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A )
5	4	intnanr 961	. . . . . . . . 9 |- -. ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear )
6	5	a1i 11	. . . . . . . 8 |- ( n e. NN -> -. ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear ) )
7	6	nrex 3000	. . . . . . 7 |- -. E. n e. NN ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear )
8	7	nex 1731	. . . . . 6 |- -. E. l E. n e. NN ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear )
9		eleq1 2689	. . . . . . . . . . . 12 |- ( x = A -> ( x e. ( EE ` n ) <-> A e. ( EE ` n ) ) )
10		neeq1 2856	. . . . . . . . . . . 12 |- ( x = A -> ( x =/= y <-> A =/= y ) )
11	9, 10	3anbi13d 1401	. . . . . . . . . . 11 |- ( x = A -> ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) <-> ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) ) )
12		opeq1 4402	. . . . . . . . . . . . 13 |- ( x = A -> <. x , y >. = <. A , y >. )
13	12	eceq1d 7783	. . . . . . . . . . . 12 |- ( x = A -> [ <. x , y >. ] `' Colinear = [ <. A , y >. ] `' Colinear )
14	13	eqeq2d 2632	. . . . . . . . . . 11 |- ( x = A -> ( l = [ <. x , y >. ] `' Colinear <-> l = [ <. A , y >. ] `' Colinear ) )
15	11, 14	anbi12d 747	. . . . . . . . . 10 |- ( x = A -> ( ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) <-> ( ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) /\ l = [ <. A , y >. ] `' Colinear ) ) )
16	15	rexbidv 3052	. . . . . . . . 9 |- ( x = A -> ( E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) <-> E. n e. NN ( ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) /\ l = [ <. A , y >. ] `' Colinear ) ) )
17	16	exbidv 1850	. . . . . . . 8 |- ( x = A -> ( E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) <-> E. l E. n e. NN ( ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) /\ l = [ <. A , y >. ] `' Colinear ) ) )
18		eleq1 2689	. . . . . . . . . . . 12 |- ( y = A -> ( y e. ( EE ` n ) <-> A e. ( EE ` n ) ) )
19		neeq2 2857	. . . . . . . . . . . 12 |- ( y = A -> ( A =/= y <-> A =/= A ) )
20	18, 19	3anbi23d 1402	. . . . . . . . . . 11 |- ( y = A -> ( ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) <-> ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) ) )
21		opeq2 4403	. . . . . . . . . . . . 13 |- ( y = A -> <. A , y >. = <. A , A >. )
22	21	eceq1d 7783	. . . . . . . . . . . 12 |- ( y = A -> [ <. A , y >. ] `' Colinear = [ <. A , A >. ] `' Colinear )
23	22	eqeq2d 2632	. . . . . . . . . . 11 |- ( y = A -> ( l = [ <. A , y >. ] `' Colinear <-> l = [ <. A , A >. ] `' Colinear ) )
24	20, 23	anbi12d 747	. . . . . . . . . 10 |- ( y = A -> ( ( ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) /\ l = [ <. A , y >. ] `' Colinear ) <-> ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear ) ) )
25	24	rexbidv 3052	. . . . . . . . 9 |- ( y = A -> ( E. n e. NN ( ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) /\ l = [ <. A , y >. ] `' Colinear ) <-> E. n e. NN ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear ) ) )
26	25	exbidv 1850	. . . . . . . 8 |- ( y = A -> ( E. l E. n e. NN ( ( A e. ( EE ` n ) /\ y e. ( EE ` n ) /\ A =/= y ) /\ l = [ <. A , y >. ] `' Colinear ) <-> E. l E. n e. NN ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear ) ) )
27	17, 26	opelopabg 4993	. . . . . . 7 |- ( ( A e. _V /\ A e. _V ) -> ( <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } <-> E. l E. n e. NN ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear ) ) )
28	27	anidms 677	. . . . . 6 |- ( A e. _V -> ( <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } <-> E. l E. n e. NN ( ( A e. ( EE ` n ) /\ A e. ( EE ` n ) /\ A =/= A ) /\ l = [ <. A , A >. ] `' Colinear ) ) )
29	8, 28	mtbiri 317	. . . . 5 |- ( A e. _V -> -. <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } )
30		elopaelxp 5191	. . . . . . 7 |- ( <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } -> <. A , A >. e. ( _V X. _V ) )
31		opelxp1 5150	. . . . . . 7 |- ( <. A , A >. e. ( _V X. _V ) -> A e. _V )
32	30, 31	syl 17	. . . . . 6 |- ( <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } -> A e. _V )
33	32	con3i 150	. . . . 5 |- ( -. A e. _V -> -. <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } )
34	29, 33	pm2.61i 176	. . . 4 |- -. <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) }
35		df-line2 32244	. . . . . . 7 |- Line = { <. <. x , y >. , l >. | E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) }
36	35	dmeqi 5325	. . . . . 6 |- dom Line = dom { <. <. x , y >. , l >. | E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) }
37		dmoprab 6741	. . . . . 6 |- dom { <. <. x , y >. , l >. | E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } = { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) }
38	36, 37	eqtri 2644	. . . . 5 |- dom Line = { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) }
39	38	eleq2i 2693	. . . 4 |- ( <. A , A >. e. dom Line <-> <. A , A >. e. { <. x , y >. | E. l E. n e. NN ( ( x e. ( EE ` n ) /\ y e. ( EE ` n ) /\ x =/= y ) /\ l = [ <. x , y >. ] `' Colinear ) } )
40	34, 39	mtbir 313	. . 3 |- -. <. A , A >. e. dom Line
41		ndmfv 6218	. . 3 |- ( -. <. A , A >. e. dom Line -> (Line ` <. A , A >. ) = (/) )
42	40, 41	ax-mp 5	. 2 |- (Line ` <. A , A >. ) = (/)
43	1, 42	eqtri 2644	1 |- ( ALine A ) = (/)

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   (/)c0 3915   <.cop 4183   {copab 4712   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ` cfv 5888  (class class class)co 6650   {coprab 6651   [cec 7740   NNcn 11020   EEcee 25768   Colinear ccolin 32144  Linecline2 32241

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-ec 7744  df-line2 32244

This theorem is referenced by: (None)