Theorem outsideofeq 25968

Description: Uniqueness law for OutsideOf. Analog of segconeq 25848. (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 957	. . . . 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 990	. . . . 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 994	. . . . 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 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 ) ) ) -> R e. ( EE ` N ) )
5		broutsideof2 25960	. . . . 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 1186	. . . 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 686	. . 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 995	. . . . 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 25960	. . . . 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 1186	. . . 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 686	. . 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 692	. 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 998	. . . . . . 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 1004	. . . . . . 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 519	. . . . . 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 453	. . . . 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 997	. . . . . 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 453	. . . . 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 992	. . . . . 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 993	. . . . . 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 740	. . . . . 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 742	. . . . . 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 25833	. . . . 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 522	. . . 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 739	. . . . . . . 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 740	. . . . . . . 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 742	. . . . . . . 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 25942	. . . . . . . . . 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 1193	. . . . . . . . 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 452	. . . . . . . 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 1282	. . . . . . 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 589	. . . . . 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 740	. . . . . . . . 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 739	. . . . . . . . 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 25864	. . . . . . . 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 742	. . . . . . . 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 25924	. . . . . . 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 589	. . . . . 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 739	. . . . . . . . . 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 740	. . . . . . . . . 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 25864	. . . . . . . . 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 742	. . . . . . . . . 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 25829	. . . . . . . . 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 25924	. . . . . . . 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 2409	. . . . . . 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 589	. . . . . 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 734	. . . . . . . . . . 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 738	. . . . . . . . . . . . 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 453	. . . . . . . . . . . 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 25829	. . . . . . . . . . 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 25924	. . . . . . . . . 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 2409	. . . . . . . . 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 599	. . . . . . . 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 734	. . . . . . . . . 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 738	. . . . . . . . . . 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 453	. . . . . . . . . 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 25924	. . . . . . . . 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 599	. . . . . . . 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 741	. . . . . . . . . 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 2650	. . . . . . . . 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 739	. . . . . . . . 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 740	. . . . . . . . 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 25939	. . . . . . . . . . 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 1193	. . . . . . . . . 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 452	. . . . . . . . 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 1282	. . . . . . . 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 371	. . . . . . 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 589	. . . . . 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 914	. . . . 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 423	. . . 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 457	. . 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 424	. 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 207	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 177   \/ wo 358   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   <.cop 3777   class class class wbr 4172   ` cfv 5413   NNcn 9956   EEcee 25731   Btwn cbtwn 25732  Cgrccgr 25733  OutsideOfcoutsideof 25957

This theorem is referenced by:  outsideofeu  25969  outsidele  25970

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-ee 25734  df-btwn 25735  df-cgr 25736  df-ofs 25821  df-ifs 25877  df-cgr3 25878  df-colinear 25879  df-fs 25880  df-outsideof 25958