Theorem btwnconn1lem13 30937

Description: Lemma for btwnconn1 30939. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)

Assertion

Ref

Expression

btwnconn1lem13

|- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( C = c \/ D = d ) )

Proof of Theorem btwnconn1lem13

Dummy variables e p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 2643	. . 3 |- ( C =/= c <-> -. C = c )
2		simp2rl 1099	. . . . . . . . . 10 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> C Btwn <. A , d >. )
3	2	adantr 472	. . . . . . . . 9 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> C Btwn <. A , d >. )
4		simp2ll 1097	. . . . . . . . . 10 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> D Btwn <. A , c >. )
5	4	adantr 472	. . . . . . . . 9 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> D Btwn <. A , c >. )
6	3, 5	jca 541	. . . . . . . 8 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( C Btwn <. A , d >. /\ D Btwn <. A , c >. ) )
7		simpl1 1033	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> N e. NN )
8		simprl1 1075	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> C e. ( EE ` N ) )
9		simpl2 1034	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> A e. ( EE ` N ) )
10		simprrl 782	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> d e. ( EE ` N ) )
11		btwncom 30852	. . . . . . . . . 10 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( C Btwn <. A , d >. <-> C Btwn <. d , A >. ) )
12	7, 8, 9, 10, 11	syl13anc 1294	. . . . . . . . 9 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( C Btwn <. A , d >. <-> C Btwn <. d , A >. ) )
13		simprl2 1076	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> D e. ( EE ` N ) )
14		simprl3 1077	. . . . . . . . . 10 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> c e. ( EE ` N ) )
15		btwncom 30852	. . . . . . . . . 10 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ A e. ( EE ` N ) /\ c e. ( EE ` N ) ) ) -> ( D Btwn <. A , c >. <-> D Btwn <. c , A >. ) )
16	7, 13, 9, 14, 15	syl13anc 1294	. . . . . . . . 9 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( D Btwn <. A , c >. <-> D Btwn <. c , A >. ) )
17	12, 16	anbi12d 725	. . . . . . . 8 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( C Btwn <. A , d >. /\ D Btwn <. A , c >. ) <-> ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) ) )
18	6, 17	syl5ib 227	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) ) )
19		axpasch 25050	. . . . . . . 8 |- ( ( N e. NN /\ ( d e. ( EE ` N ) /\ c e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) )
20	7, 10, 14, 9, 8, 13, 19	syl132anc 1310	. . . . . . 7 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( C Btwn <. d , A >. /\ D Btwn <. c , A >. ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) )
21	18, 20	syld 44	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) )
22	21	imp 436	. . . . 5 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) ) -> E. e e. ( EE ` N ) ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) )
23		simpll1 1069	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> N e. NN )
24	14	adantr 472	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> c e. ( EE ` N ) )
25	8	adantr 472	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> C e. ( EE ` N ) )
26	10	adantr 472	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> d e. ( EE ` N ) )
27		axsegcon 25036	. . . . . . . . . 10 |- ( ( N e. NN /\ ( c e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> E. p e. ( EE ` N ) ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) )
28	23, 24, 25, 25, 26, 27	syl122anc 1301	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> E. p e. ( EE ` N ) ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) )
29		simpr 468	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> e e. ( EE ` N ) )
30		axsegcon 25036	. . . . . . . . . 10 |- ( ( N e. NN /\ ( d e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) -> E. r e. ( EE ` N ) ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) )
31	23, 26, 25, 25, 29, 30	syl122anc 1301	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> E. r e. ( EE ` N ) ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) )
32		reeanv 2944	. . . . . . . . 9 |- ( E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) <-> ( E. p e. ( EE ` N ) ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ E. r e. ( EE ` N ) ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) )
33	28, 31, 32	sylanbrc 677	. . . . . . . 8 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) )
34	33	adantr 472	. . . . . . 7 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) )
35	7	ad2antrr 740	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> N e. NN )
36		simprl 772	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> p e. ( EE ` N ) )
37		simprr 774	. . . . . . . . . . . . 13 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> r e. ( EE ` N ) )
38		axsegcon 25036	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) /\ ( r e. ( EE ` N ) /\ p e. ( EE ` N ) ) ) -> E. q e. ( EE ` N ) ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
39	35, 36, 37, 37, 36, 38	syl122anc 1301	. . . . . . . . . . . 12 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) -> E. q e. ( EE ` N ) ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
40	39	adantr 472	. . . . . . . . . . 11 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) -> E. q e. ( EE ` N ) ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
41		simp-4l 784	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
42		simplrl 778	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) )
43	42	ad2antrr 740	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) )
44	10	ad3antrrr 744	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> d e. ( EE ` N ) )
45		simprrr 783	. . . . . . . . . . . . . . . . 17 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) -> b e. ( EE ` N ) )
46	45	ad3antrrr 744	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> b e. ( EE ` N ) )
47		simpllr 777	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> e e. ( EE ` N ) )
48	44, 46, 47	3jca 1210	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) )
49	43, 48	jca 541	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) )
50		simplrl 778	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> p e. ( EE ` N ) )
51		simpr 468	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> q e. ( EE ` N ) )
52		simplrr 779	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> r e. ( EE ` N ) )
53	50, 51, 52	3jca 1210	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( p e. ( EE ` N ) /\ q e. ( EE ` N ) /\ r e. ( EE ` N ) ) )
54	41, 49, 53	3jca 1210	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) -> ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) /\ ( p e. ( EE ` N ) /\ q e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) )
55		simp1ll 1093	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> A =/= B )
56	55	ad3antrrr 744	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) -> A =/= B )
57	56	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> A =/= B )
58		simp1lr 1094	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> B =/= C )
59	58	ad3antrrr 744	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) -> B =/= C )
60	59	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> B =/= C )
61		simpllr 777	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) -> C =/= c )
62	61	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> C =/= c )
63	57, 60, 62	3jca 1210	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( A =/= B /\ B =/= C /\ C =/= c ) )
64		simpl1r 1082	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) )
65	64	ad3antrrr 744	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) )
66	63, 65	jca 541	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) )
67		simpll2 1070	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) -> ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
68	67	ad2antrr 740	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) )
69		simpl3l 1085	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) )
70	69	ad3antrrr 744	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) )
71		simpl3r 1086	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) -> ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) )
72	71	ad3antrrr 744	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) )
73	70, 72	jca 541	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) )
74	66, 68, 73	3jca 1210	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) )
75		simpllr 777	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) )
76		simplrl 778	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) )
77		simplrr 779	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) )
78		simpr 468	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) )
79	76, 77, 78	3jca 1210	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) )
80	74, 75, 79	jca32 544	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) -> ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) ) )
81		btwnconn1lem12 30936	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ e e. ( EE ` N ) ) ) /\ ( p e. ( EE ` N ) /\ q e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) ) ) -> D = d )
82	54, 80, 81	syl2an 485	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ q e. ( EE ` N ) ) /\ ( ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) -> D = d )
83	82	an4s 842	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) /\ ( q e. ( EE ` N ) /\ ( r Btwn <. p , q >. /\ <. r , q >.Cgr <. r , p >. ) ) ) -> D = d )
84	40, 83	rexlimddv 2875	. . . . . . . . . 10 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) ) /\ ( ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) -> D = d )
85	84	an4s 842	. . . . . . . . 9 |- ( ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) /\ ( ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) /\ ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) ) ) -> D = d )
86	85	exp32 616	. . . . . . . 8 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> ( ( p e. ( EE ` N ) /\ r e. ( EE ` N ) ) -> ( ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) -> D = d ) ) )
87	86	rexlimdvv 2877	. . . . . . 7 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> ( E. p e. ( EE ` N ) E. r e. ( EE ` N ) ( ( C Btwn <. c , p >. /\ <. C , p >.Cgr <. C , d >. ) /\ ( C Btwn <. d , r >. /\ <. C , r >.Cgr <. C , e >. ) ) -> D = d ) )
88	34, 87	mpd 15	. . . . . 6 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ e e. ( EE ` N ) ) /\ ( ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> D = d )
89	88	an4s 842	. . . . 5 |- ( ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) ) /\ ( e e. ( EE ` N ) /\ ( e Btwn <. C , c >. /\ e Btwn <. D , d >. ) ) ) -> D = d )
90	22, 89	rexlimddv 2875	. . . 4 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ C =/= c ) ) -> D = d )
91	90	expr 626	. . 3 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( C =/= c -> D = d ) )
92	1, 91	syl5bir 226	. 2 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( -. C = c -> D = d ) )
93	92	orrd 385	1 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) ) ) ) /\ ( ( ( A =/= B /\ B =/= C ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) ) -> ( C = c \/ D = d ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   <.cop 3965   class class class wbr 4395   ` cfv 5589   NNcn 10631   EEcee 24997   Btwn cbtwn 24998  Cgrccgr 24999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-ee 25000  df-btwn 25001  df-cgr 25002  df-ofs 30821  df-colinear 30877  df-ifs 30878  df-cgr3 30879  df-fs 30880

This theorem is referenced by:  btwnconn1lem14  30938