Theorem outsideofeq 29207

Description: Uniqueness law for OutsideOf. Analog of segconeq 29087. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
outsideofeq	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )

Proof of Theorem outsideofeq

Step	Hyp	Ref	Expression
1		simp1 991	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> N e. NN )
2		simp21 1024	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
3		simp32 1028	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> X e. ( EE ` N ) )
4		simp22 1025	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> R e. ( EE ` N ) )
5		broutsideof2 29199	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ X e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( AOutsideOf <. X , R >. <-> ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) ) )
6	1, 2, 3, 4, 5	syl13anc 1225	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( AOutsideOf <. X , R >. <-> ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) ) )
7	6	anbi1d 704	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) <-> ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) ) )
8		simp33 1029	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> Y e. ( EE ` N ) )
9		broutsideof2 29199	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ Y e. ( EE ` N ) /\ R e. ( EE ` N ) ) ) -> ( AOutsideOf <. Y , R >. <-> ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) ) )
10	1, 2, 8, 4, 9	syl13anc 1225	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( AOutsideOf <. Y , R >. <-> ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) ) )
11	10	anbi1d 704	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) <-> ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) )
12	7, 11	anbi12d 710	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) <-> ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) )
13		simpll3 1032	. . . . . . 7 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) )
14		simprl3 1038	. . . . . . 7 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) )
15	13, 14	jca 532	. . . . . 6 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) )
16	15	adantl 466	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) )
17		simpll2 1031	. . . . . 6 |- ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> R =/= A )
18	17	adantl 466	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> R =/= A )
19		simp23 1026	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
20		simp31 1027	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
21		simprlr 762	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. B , C >. )
22		simprrr 764	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , Y >.Cgr <. B , C >. )
23	1, 2, 3, 2, 8, 19, 20, 21, 22	cgrtr3and 29072	. . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
24	16, 18, 23	jca32 535	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) )
25		simprll 761	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , R >. )
26		simprlr 762	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , R >. )
27		simprrr 764	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
28		midofsegid 29181	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
29	1, 2, 4, 3, 8, 28	syl122anc 1232	. . . . . . . . 9 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
30	29	adantr 465	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) )
31	25, 26, 27, 30	mp3and 1322	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
32	31	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ Y Btwn <. A , R >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
33		simprlr 762	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , R >. )
34		simprll 761	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , X >. )
35	1, 2, 8, 4, 3, 33, 34	btwnexchand 29103	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y Btwn <. A , X >. )
36		simprrr 764	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
37	1, 2, 3, 8, 35, 36	endofsegidand 29163	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
38	37	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( R Btwn <. A , X >. /\ Y Btwn <. A , R >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
39		simprll 761	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , R >. )
40		simprlr 762	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , Y >. )
41	1, 2, 3, 4, 8, 39, 40	btwnexchand 29103	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X Btwn <. A , Y >. )
42		simprrr 764	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , X >.Cgr <. A , Y >. )
43	1, 2, 3, 2, 8, 42	cgrcomand 29068	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> <. A , Y >.Cgr <. A , X >. )
44	1, 2, 8, 3, 41, 43	endofsegidand 29163	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> Y = X )
45	44	eqcomd 2468	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
46	45	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( X Btwn <. A , R >. /\ R Btwn <. A , Y >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
47		simprr 756	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> X Btwn <. A , Y >. )
48		simplrr 760	. . . . . . . . . . . . 13 |- ( ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) -> <. A , X >.Cgr <. A , Y >. )
49	48	adantl 466	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> <. A , X >.Cgr <. A , Y >. )
50	1, 2, 3, 2, 8, 49	cgrcomand 29068	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> <. A , Y >.Cgr <. A , X >. )
51	1, 2, 8, 3, 47, 50	endofsegidand 29163	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> Y = X )
52	51	eqcomd 2468	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ X Btwn <. A , Y >. ) ) -> X = Y )
53	52	expr 615	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( X Btwn <. A , Y >. -> X = Y ) )
54		simprr 756	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> Y Btwn <. A , X >. )
55		simplrr 760	. . . . . . . . . . 11 |- ( ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) -> <. A , X >.Cgr <. A , Y >. )
56	55	adantl 466	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> <. A , X >.Cgr <. A , Y >. )
57	1, 2, 3, 8, 54, 56	endofsegidand 29163	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) /\ Y Btwn <. A , X >. ) ) -> X = Y )
58	57	expr 615	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( Y Btwn <. A , X >. -> X = Y ) )
59		simprrl 763	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R =/= A )
60	59	necomd 2731	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> A =/= R )
61		simprll 761	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , X >. )
62		simprlr 762	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> R Btwn <. A , Y >. )
63		btwnconn1 29178	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) ) /\ ( X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
64	1, 2, 4, 3, 8, 63	syl122anc 1232	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
65	64	adantr 465	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( ( A =/= R /\ R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) ) )
66	60, 61, 62, 65	mp3and 1322	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> ( X Btwn <. A , Y >. \/ Y Btwn <. A , X >. ) )
67	53, 58, 66	mpjaod 381	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
68	67	exp32 605	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( R Btwn <. A , X >. /\ R Btwn <. A , Y >. ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
69	32, 38, 46, 68	ccased 940	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) -> ( ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) -> X = Y ) ) )
70	69	imp32 433	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ ( R =/= A /\ <. A , X >.Cgr <. A , Y >. ) ) ) -> X = Y )
71	24, 70	syldan 470	. . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) /\ ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) ) -> X = Y )
72	71	ex 434	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( ( X =/= A /\ R =/= A /\ ( X Btwn <. A , R >. \/ R Btwn <. A , X >. ) ) /\ <. A , X >.Cgr <. B , C >. ) /\ ( ( Y =/= A /\ R =/= A /\ ( Y Btwn <. A , R >. \/ R Btwn <. A , Y >. ) ) /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )
73	12, 72	sylbid 215	1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ R e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ X e. ( EE ` N ) /\ Y e. ( EE ` N ) ) ) -> ( ( ( AOutsideOf <. X , R >. /\ <. A , X >.Cgr <. B , C >. ) /\ ( AOutsideOf <. Y , R >. /\ <. A , Y >.Cgr <. B , C >. ) ) -> X = Y ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   =/= wne 2655   <.cop 4026   class class class wbr 4440   ` cfv 5579   NNcn 10525   EEcee 23860   Btwn cbtwn 23861  Cgrccgr 23862  OutsideOfcoutsideof 29196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-ee 23863  df-btwn 23864  df-cgr 23865  df-ofs 29060  df-colinear 29116  df-ifs 29117  df-cgr3 29118  df-fs 29119  df-outsideof 29197

This theorem is referenced by:  outsideofeu  29208  outsidele  29209