Theorem broutsideof2 32229

Description: Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
broutsideof2	|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( POutsideOf <. A , B >. <-> ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )

Proof of Theorem broutsideof2

Step	Hyp	Ref	Expression
1		broutsideof 32228	. 2 |- ( POutsideOf <. A , B >. <-> ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) )
2		btwntriv1 32123	. . . . . . . . 9 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A Btwn <. A , B >. )
3	2	3adant3r1 1274	. . . . . . . 8 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> A Btwn <. A , B >. )
4		breq1 4656	. . . . . . . 8 |- ( A = P -> ( A Btwn <. A , B >. <-> P Btwn <. A , B >. ) )
5	3, 4	syl5ibcom 235	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A = P -> P Btwn <. A , B >. ) )
6	5	necon3bd 2808	. . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( -. P Btwn <. A , B >. -> A =/= P ) )
7	6	imp 445	. . . . 5 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ -. P Btwn <. A , B >. ) -> A =/= P )
8	7	adantrl 752	. . . 4 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) ) -> A =/= P )
9		btwntriv2 32119	. . . . . . . . 9 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> B Btwn <. A , B >. )
10	9	3adant3r1 1274	. . . . . . . 8 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> B Btwn <. A , B >. )
11		breq1 4656	. . . . . . . 8 |- ( B = P -> ( B Btwn <. A , B >. <-> P Btwn <. A , B >. ) )
12	10, 11	syl5ibcom 235	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( B = P -> P Btwn <. A , B >. ) )
13	12	necon3bd 2808	. . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( -. P Btwn <. A , B >. -> B =/= P ) )
14	13	imp 445	. . . . 5 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ -. P Btwn <. A , B >. ) -> B =/= P )
15	14	adantrl 752	. . . 4 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) ) -> B =/= P )
16		brcolinear 32166	. . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( P Colinear <. A , B >. <-> ( P Btwn <. A , B >. \/ A Btwn <. B , P >. \/ B Btwn <. P , A >. ) ) )
17		pm2.24 121	. . . . . . . 8 |- ( P Btwn <. A , B >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) )
18	17	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( P Btwn <. A , B >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )
19		3anrot 1043	. . . . . . . . . 10 |- ( ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) <-> ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ P e. ( EE ` N ) ) )
20		btwncom 32121	. . . . . . . . . 10 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ P e. ( EE ` N ) ) ) -> ( A Btwn <. B , P >. <-> A Btwn <. P , B >. ) )
21	19, 20	sylan2b 492	. . . . . . . . 9 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Btwn <. B , P >. <-> A Btwn <. P , B >. ) )
22		orc 400	. . . . . . . . 9 |- ( A Btwn <. P , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) )
23	21, 22	syl6bi 243	. . . . . . . 8 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Btwn <. B , P >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) )
24	23	a1dd 50	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Btwn <. B , P >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )
25		olc 399	. . . . . . . . 9 |- ( B Btwn <. P , A >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) )
26	25	a1d 25	. . . . . . . 8 |- ( B Btwn <. P , A >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) )
27	26	a1i 11	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( B Btwn <. P , A >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )
28	18, 24, 27	3jaod 1392	. . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( P Btwn <. A , B >. \/ A Btwn <. B , P >. \/ B Btwn <. P , A >. ) -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )
29	16, 28	sylbid 230	. . . . 5 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( P Colinear <. A , B >. -> ( -. P Btwn <. A , B >. -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )
30	29	imp32 449	. . . 4 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) ) -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) )
31	8, 15, 30	3jca 1242	. . 3 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) ) -> ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) )
32		simp3 1063	. . . . . 6 |- ( ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) -> ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) )
33		3ancomb 1047	. . . . . . . 8 |- ( ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) <-> ( P e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) )
34		btwncolinear2 32177	. . . . . . . 8 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( A Btwn <. P , B >. -> P Colinear <. A , B >. ) )
35	33, 34	sylan2b 492	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Btwn <. P , B >. -> P Colinear <. A , B >. ) )
36		btwncolinear1 32176	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( B Btwn <. P , A >. -> P Colinear <. A , B >. ) )
37	35, 36	jaod 395	. . . . . 6 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) -> P Colinear <. A , B >. ) )
38	32, 37	syl5 34	. . . . 5 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) -> P Colinear <. A , B >. ) )
39	38	imp 445	. . . 4 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) -> P Colinear <. A , B >. )
40		simpr2 1068	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) ) -> A =/= P )
41	40	neneqd 2799	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) ) -> -. A = P )
42		simprl1 1106	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> A Btwn <. P , B >. )
43		simprr 796	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> P Btwn <. A , B >. )
44		simpl 473	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> N e. NN )
45		simpr2 1068	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
46		simpr1 1067	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> P e. ( EE ` N ) )
47		simpr3 1069	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
48		btwnswapid 32124	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ P e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( A Btwn <. P , B >. /\ P Btwn <. A , B >. ) -> A = P ) )
49	44, 45, 46, 47, 48	syl13anc 1328	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( A Btwn <. P , B >. /\ P Btwn <. A , B >. ) -> A = P ) )
50	49	adantr 481	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> ( ( A Btwn <. P , B >. /\ P Btwn <. A , B >. ) -> A = P ) )
51	42, 43, 50	mp2and 715	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> A = P )
52	51	expr 643	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) ) -> ( P Btwn <. A , B >. -> A = P ) )
53	41, 52	mtod 189	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( A Btwn <. P , B >. /\ A =/= P /\ B =/= P ) ) -> -. P Btwn <. A , B >. )
54	53	3exp2 1285	. . . . . . . 8 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A Btwn <. P , B >. -> ( A =/= P -> ( B =/= P -> -. P Btwn <. A , B >. ) ) ) )
55		simpr3 1069	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) ) -> B =/= P )
56	55	neneqd 2799	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) ) -> -. B = P )
57		simprl1 1106	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> B Btwn <. P , A >. )
58		simprr 796	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> P Btwn <. A , B >. )
59	44, 46, 45, 47, 58	btwncomand 32122	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> P Btwn <. B , A >. )
60		3anrot 1043	. . . . . . . . . . . . . 14 |- ( ( B e. ( EE ` N ) /\ P e. ( EE ` N ) /\ A e. ( EE ` N ) ) <-> ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) )
61		btwnswapid 32124	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ P e. ( EE ` N ) /\ A e. ( EE ` N ) ) ) -> ( ( B Btwn <. P , A >. /\ P Btwn <. B , A >. ) -> B = P ) )
62	60, 61	sylan2br 493	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( B Btwn <. P , A >. /\ P Btwn <. B , A >. ) -> B = P ) )
63	62	adantr 481	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> ( ( B Btwn <. P , A >. /\ P Btwn <. B , A >. ) -> B = P ) )
64	57, 59, 63	mp2and 715	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) /\ P Btwn <. A , B >. ) ) -> B = P )
65	64	expr 643	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) ) -> ( P Btwn <. A , B >. -> B = P ) )
66	56, 65	mtod 189	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( B Btwn <. P , A >. /\ A =/= P /\ B =/= P ) ) -> -. P Btwn <. A , B >. )
67	66	3exp2 1285	. . . . . . . 8 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( B Btwn <. P , A >. -> ( A =/= P -> ( B =/= P -> -. P Btwn <. A , B >. ) ) ) )
68	54, 67	jaod 395	. . . . . . 7 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) -> ( A =/= P -> ( B =/= P -> -. P Btwn <. A , B >. ) ) ) )
69	68	com12 32	. . . . . 6 |- ( ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) -> ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A =/= P -> ( B =/= P -> -. P Btwn <. A , B >. ) ) ) )
70	69	com4l 92	. . . . 5 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( A =/= P -> ( B =/= P -> ( ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) -> -. P Btwn <. A , B >. ) ) ) )
71	70	3imp2 1282	. . . 4 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) -> -. P Btwn <. A , B >. )
72	39, 71	jca 554	. . 3 |- ( ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) /\ ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) -> ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) )
73	31, 72	impbida 877	. 2 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( ( P Colinear <. A , B >. /\ -. P Btwn <. A , B >. ) <-> ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )
74	1, 73	syl5bb 272	1 |- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( POutsideOf <. A , B >. <-> ( A =/= P /\ B =/= P /\ ( A Btwn <. P , B >. \/ B Btwn <. P , A >. ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769   Colinear ccolin 32144  OutsideOfcoutsideof 32226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-colinear 32146  df-outsideof 32227

This theorem is referenced by:  outsidene1  32230  outsidene2  32231  btwnoutside  32232  broutsideof3  32233  outsideofcom  32235  outsideoftr  32236  outsideofeq  32237  outsideofeu  32238  lineunray  32254