Theorem btwnconn1lem13 30424

Description: Lemma for btwnconn1 30426. 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 2600	. . 3 |- ( C =/= c <-> -. C = c )
2		simp2rl 1066	. . . . . . . . . 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 463	. . . . . . . . 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 1064	. . . . . . . . . 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 463	. . . . . . . . 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 530	. . . . . . . 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 1000	. . . . . . . . . 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 1042	. . . . . . . . . 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 1001	. . . . . . . . . 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 766	. . . . . . . . . 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 30339	. . . . . . . . . 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 1232	. . . . . . . . 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 1043	. . . . . . . . . 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 1044	. . . . . . . . . 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 30339	. . . . . . . . . 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 1232	. . . . . . . . 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 709	. . . . . . . 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 219	. . . . . . 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 24648	. . . . . . . 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 1248	. . . . . . 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 42	. . . . . 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 427	. . . . 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 1036	. . . . . . . . . 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 463	. . . . . . . . . 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 463	. . . . . . . . . 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 463	. . . . . . . . . 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 24634	. . . . . . . . . 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 1239	. . . . . . . . 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 459	. . . . . . . . . 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 24634	. . . . . . . . . 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 1239	. . . . . . . . 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 2974	. . . . . . . . 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 662	. . . . . . . 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 463	. . . . . . 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 724	. . . . . . . . . . . . 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 756	. . . . . . . . . . . . 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 758	. . . . . . . . . . . . 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 24634	. . . . . . . . . . . . 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 1239	. . . . . . . . . . . 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 463	. . . . . . . . . . 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 768	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 761	. . . . . . . . . . . . . . . 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 1177	. . . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . 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 1177	. . . . . . . . . . . . . 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 1177	. . . . . . . . . . . . 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 1060	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . 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 1061	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . 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 761	. . . . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . . . . 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 1177	. . . . . . . . . . . . . . . 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 1049	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . . 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 1037	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . 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 1052	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 1053	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . . 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 1177	. . . . . . . . . . . . . 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 761	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . 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 1177	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . . 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 30423	. . . . . . . . . . . . 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 475	. . . . . . . . . . . 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 827	. . . . . . . . . . 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 2899	. . . . . . . . . 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 827	. . . . . . . . 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 603	. . . . . . . 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 2901	. . . . . . 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 827	. . . . 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 2899	. . . 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 613	. . 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 218	. 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 376	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 184   \/ wo 366   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   E.wrex 2754   <.cop 3977   class class class wbr 4394   ` cfv 5568   NNcn 10575   EEcee 24595   Btwn cbtwn 24596  Cgrccgr 24597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-ee 24598  df-btwn 24599  df-cgr 24600  df-ofs 30308  df-colinear 30364  df-ifs 30365  df-cgr3 30366  df-fs 30367

This theorem is referenced by:  btwnconn1lem14  30425