Theorem btwnconn1lem13 28130

Description: Lemma for btwnconn1 28132. 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 2608	. . 3 |- ( C =/= c <-> -. C = c )
2		simp2rl 1057	. . . . . . . . . 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 465	. . . . . . . . 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 1055	. . . . . . . . . 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 465	. . . . . . . . 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 532	. . . . . . . 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 991	. . . . . . . . . 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 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 ) ) ) ) -> C e. ( EE ` N ) )
9		simpl2 992	. . . . . . . . . 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 763	. . . . . . . . . 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 28045	. . . . . . . . . 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 1220	. . . . . . . . 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 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 ) ) ) ) -> D e. ( EE ` N ) )
14		simprl3 1035	. . . . . . . . . 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 28045	. . . . . . . . . 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 1220	. . . . . . . . 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 710	. . . . . . . 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 23187	. . . . . . . 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 1236	. . . . . . 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 429	. . . . 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 1027	. . . . . . . . . 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 465	. . . . . . . . . 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 465	. . . . . . . . . 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 465	. . . . . . . . . 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 23173	. . . . . . . . . 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 1227	. . . . . . . . 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 461	. . . . . . . . . 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 23173	. . . . . . . . . 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 1227	. . . . . . . . 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 2888	. . . . . . . . 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 664	. . . . . . . 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 465	. . . . . . 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 725	. . . . . . . . . . . . 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 755	. . . . . . . . . . . . 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 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 ) ) ) -> r e. ( EE ` N ) )
38		axsegcon 23173	. . . . . . . . . . . . 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 1227	. . . . . . . . . . . 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 465	. . . . . . . . . . 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 765	. . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 764	. . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . 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 1051	. . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . 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 1052	. . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . . . 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 1040	. . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . . 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 1028	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . . 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 535	. . . . . . . . . . . . 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 28129	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 822	. . . . . . . . . . 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 2845	. . . . . . . . . 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 822	. . . . . . . . 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 605	. . . . . . . 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 2847	. . . . . . 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 822	. . . . 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 2845	. . . 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 615	. . 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 378	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 368   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2606   E.wrex 2716   <.cop 3883   class class class wbr 4292   ` cfv 5418   NNcn 10322   EEcee 23134   Btwn cbtwn 23135  Cgrccgr 23136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-ee 23137  df-btwn 23138  df-cgr 23139  df-ofs 28014  df-colinear 28070  df-ifs 28071  df-cgr3 28072  df-fs 28073

This theorem is referenced by:  btwnconn1lem14  28131