Theorem nb3graprlem2 24579

Description: Lemma 2 for nb3grapr 24580. (Contributed by Alexander van der Vekens, 17-Oct-2017.)

Assertion

Ref	Expression
nb3graprlem2	|- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> E. v e. V E. w e. ( V \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } ) )

Distinct variable groups:   v, E, w   v, V, w   v, A, w   v, B, w   v, C, w

Allowed substitution hints:   X( w, v)   Y( w, v)   Z( w, v)

Proof of Theorem nb3graprlem2

Step	Hyp	Ref	Expression
1		sneq 4042	. . . . . 6 |- ( v = A -> { v } = { A } )
2	1	difeq2d 3618	. . . . 5 |- ( v = A -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { A } ) )
3		preq1 4111	. . . . . 6 |- ( v = A -> { v , w } = { A , w } )
4	3	eqeq2d 2471	. . . . 5 |- ( v = A -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { A , w } ) )
5	2, 4	rexeqbidv 3069	. . . 4 |- ( v = A -> ( E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> E. w e. ( { A , B , C } \ { A } ) ( <. V , E >. Neighbors A ) = { A , w } ) )
6		sneq 4042	. . . . . 6 |- ( v = B -> { v } = { B } )
7	6	difeq2d 3618	. . . . 5 |- ( v = B -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { B } ) )
8		preq1 4111	. . . . . 6 |- ( v = B -> { v , w } = { B , w } )
9	8	eqeq2d 2471	. . . . 5 |- ( v = B -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { B , w } ) )
10	7, 9	rexeqbidv 3069	. . . 4 |- ( v = B -> ( E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> E. w e. ( { A , B , C } \ { B } ) ( <. V , E >. Neighbors A ) = { B , w } ) )
11		sneq 4042	. . . . . 6 |- ( v = C -> { v } = { C } )
12	11	difeq2d 3618	. . . . 5 |- ( v = C -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { C } ) )
13		preq1 4111	. . . . . 6 |- ( v = C -> { v , w } = { C , w } )
14	13	eqeq2d 2471	. . . . 5 |- ( v = C -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { C , w } ) )
15	12, 14	rexeqbidv 3069	. . . 4 |- ( v = C -> ( E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> E. w e. ( { A , B , C } \ { C } ) ( <. V , E >. Neighbors A ) = { C , w } ) )
16	5, 10, 15	rextpg 4084	. . 3 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( E. v e. { A , B , C } E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> ( E. w e. ( { A , B , C } \ { A } ) ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. ( { A , B , C } \ { B } ) ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. ( { A , B , C } \ { C } ) ( <. V , E >. Neighbors A ) = { C , w } ) ) )
17	16	3ad2ant1 1017	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( E. v e. { A , B , C } E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> ( E. w e. ( { A , B , C } \ { A } ) ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. ( { A , B , C } \ { B } ) ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. ( { A , B , C } \ { C } ) ( <. V , E >. Neighbors A ) = { C , w } ) ) )
18		simpl 457	. . . 4 |- ( ( V = { A , B , C } /\ V USGrph E ) -> V = { A , B , C } )
19		difeq1 3611	. . . . . 6 |- ( V = { A , B , C } -> ( V \ { v } ) = ( { A , B , C } \ { v } ) )
20	19	adantr 465	. . . . 5 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V \ { v } ) = ( { A , B , C } \ { v } ) )
21	20	rexeqdv 3061	. . . 4 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( E. w e. ( V \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } ) )
22	18, 21	rexeqbidv 3069	. . 3 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( E. v e. V E. w e. ( V \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> E. v e. { A , B , C } E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } ) )
23	22	3ad2ant2 1018	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( E. v e. V E. w e. ( V \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } <-> E. v e. { A , B , C } E. w e. ( { A , B , C } \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } ) )
24		preq2 4112	. . . . . . . 8 |- ( w = B -> { A , w } = { A , B } )
25	24	eqeq2d 2471	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , B } ) )
26		preq2 4112	. . . . . . . 8 |- ( w = C -> { A , w } = { A , C } )
27	26	eqeq2d 2471	. . . . . . 7 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , C } ) )
28	25, 27	rexprg 4082	. . . . . 6 |- ( ( B e. Y /\ C e. Z ) -> ( E. w e. { B , C } ( <. V , E >. Neighbors A ) = { A , w } <-> ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) ) )
29	28	3adant1 1014	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( E. w e. { B , C } ( <. V , E >. Neighbors A ) = { A , w } <-> ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) ) )
30		preq2 4112	. . . . . . . . 9 |- ( w = C -> { B , w } = { B , C } )
31	30	eqeq2d 2471	. . . . . . . 8 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , C } ) )
32		preq2 4112	. . . . . . . . 9 |- ( w = A -> { B , w } = { B , A } )
33	32	eqeq2d 2471	. . . . . . . 8 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , A } ) )
34	31, 33	rexprg 4082	. . . . . . 7 |- ( ( C e. Z /\ A e. X ) -> ( E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) ) )
35	34	ancoms 453	. . . . . 6 |- ( ( A e. X /\ C e. Z ) -> ( E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) ) )
36	35	3adant2 1015	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) ) )
37		preq2 4112	. . . . . . . 8 |- ( w = A -> { C , w } = { C , A } )
38	37	eqeq2d 2471	. . . . . . 7 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , A } ) )
39		preq2 4112	. . . . . . . 8 |- ( w = B -> { C , w } = { C , B } )
40	39	eqeq2d 2471	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , B } ) )
41	38, 40	rexprg 4082	. . . . . 6 |- ( ( A e. X /\ B e. Y ) -> ( E. w e. { A , B } ( <. V , E >. Neighbors A ) = { C , w } <-> ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) )
42	41	3adant3 1016	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( E. w e. { A , B } ( <. V , E >. Neighbors A ) = { C , w } <-> ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) )
43	29, 36, 42	3orbi123d 1298	. . . 4 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( E. w e. { B , C } ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. { A , B } ( <. V , E >. Neighbors A ) = { C , w } ) <-> ( ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) \/ ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) \/ ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) ) )
44	43	3ad2ant1 1017	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( E. w e. { B , C } ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. { A , B } ( <. V , E >. Neighbors A ) = { C , w } ) <-> ( ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) \/ ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) \/ ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) ) )
45		tprot 4127	. . . . . . . . 9 |- { A , B , C } = { B , C , A }
46	45	a1i 11	. . . . . . . 8 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> { A , B , C } = { B , C , A } )
47	46	difeq1d 3617	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = ( { B , C , A } \ { A } ) )
48		necom 2726	. . . . . . . . 9 |- ( A =/= B <-> B =/= A )
49		necom 2726	. . . . . . . . 9 |- ( A =/= C <-> C =/= A )
50		diftpsn3 4170	. . . . . . . . 9 |- ( ( B =/= A /\ C =/= A ) -> ( { B , C , A } \ { A } ) = { B , C } )
51	48, 49, 50	syl2anb 479	. . . . . . . 8 |- ( ( A =/= B /\ A =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
52	51	3adant3 1016	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
53	47, 52	eqtrd 2498	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = { B , C } )
54	53	rexeqdv 3061	. . . . 5 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( E. w e. ( { A , B , C } \ { A } ) ( <. V , E >. Neighbors A ) = { A , w } <-> E. w e. { B , C } ( <. V , E >. Neighbors A ) = { A , w } ) )
55		tprot 4127	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
56	55	eqcomi 2470	. . . . . . . . 9 |- { A , B , C } = { C , A , B }
57	56	a1i 11	. . . . . . . 8 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> { A , B , C } = { C , A , B } )
58	57	difeq1d 3617	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = ( { C , A , B } \ { B } ) )
59		necom 2726	. . . . . . . . . . . 12 |- ( B =/= C <-> C =/= B )
60	59	anbi1i 695	. . . . . . . . . . 11 |- ( ( B =/= C /\ A =/= B ) <-> ( C =/= B /\ A =/= B ) )
61	60	biimpi 194	. . . . . . . . . 10 |- ( ( B =/= C /\ A =/= B ) -> ( C =/= B /\ A =/= B ) )
62	61	ancoms 453	. . . . . . . . 9 |- ( ( A =/= B /\ B =/= C ) -> ( C =/= B /\ A =/= B ) )
63		diftpsn3 4170	. . . . . . . . 9 |- ( ( C =/= B /\ A =/= B ) -> ( { C , A , B } \ { B } ) = { C , A } )
64	62, 63	syl 16	. . . . . . . 8 |- ( ( A =/= B /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
65	64	3adant2 1015	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
66	58, 65	eqtrd 2498	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = { C , A } )
67	66	rexeqdv 3061	. . . . 5 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( E. w e. ( { A , B , C } \ { B } ) ( <. V , E >. Neighbors A ) = { B , w } <-> E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } ) )
68		diftpsn3 4170	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
69	68	3adant1 1014	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
70	69	rexeqdv 3061	. . . . 5 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( E. w e. ( { A , B , C } \ { C } ) ( <. V , E >. Neighbors A ) = { C , w } <-> E. w e. { A , B } ( <. V , E >. Neighbors A ) = { C , w } ) )
71	54, 67, 70	3orbi123d 1298	. . . 4 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( ( E. w e. ( { A , B , C } \ { A } ) ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. ( { A , B , C } \ { B } ) ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. ( { A , B , C } \ { C } ) ( <. V , E >. Neighbors A ) = { C , w } ) <-> ( E. w e. { B , C } ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. { A , B } ( <. V , E >. Neighbors A ) = { C , w } ) ) )
72	71	3ad2ant3 1019	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( E. w e. ( { A , B , C } \ { A } ) ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. ( { A , B , C } \ { B } ) ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. ( { A , B , C } \ { C } ) ( <. V , E >. Neighbors A ) = { C , w } ) <-> ( E. w e. { B , C } ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. { C , A } ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. { A , B } ( <. V , E >. Neighbors A ) = { C , w } ) ) )
73		prcom 4110	. . . . . . . 8 |- { C , B } = { B , C }
74	73	eqeq2i 2475	. . . . . . 7 |- ( ( <. V , E >. Neighbors A ) = { C , B } <-> ( <. V , E >. Neighbors A ) = { B , C } )
75	74	orbi2i 519	. . . . . 6 |- ( ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { C , B } ) <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , C } ) )
76		oridm 514	. . . . . 6 |- ( ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , C } ) <-> ( <. V , E >. Neighbors A ) = { B , C } )
77	75, 76	bitr2i 250	. . . . 5 |- ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { C , B } ) )
78	77	a1i 11	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) )
79		nbgranself2 24563	. . . . . . . . . 10 |- ( V USGrph E -> A e/ ( <. V , E >. Neighbors A ) )
80		df-nel 2655	. . . . . . . . . . 11 |- ( A e/ ( <. V , E >. Neighbors A ) <-> -. A e. ( <. V , E >. Neighbors A ) )
81		prid2g 4139	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { B , A } )
82	81	3ad2ant1 1017	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { B , A } )
83		eleq2 2530	. . . . . . . . . . . . 13 |- ( ( <. V , E >. Neighbors A ) = { B , A } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { B , A } ) )
84	82, 83	syl5ibrcom 222	. . . . . . . . . . . 12 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( <. V , E >. Neighbors A ) = { B , A } -> A e. ( <. V , E >. Neighbors A ) ) )
85	84	con3rr3 136	. . . . . . . . . . 11 |- ( -. A e. ( <. V , E >. Neighbors A ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { B , A } ) )
86	80, 85	sylbi 195	. . . . . . . . . 10 |- ( A e/ ( <. V , E >. Neighbors A ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { B , A } ) )
87	79, 86	syl 16	. . . . . . . . 9 |- ( V USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { B , A } ) )
88	87	adantl 466	. . . . . . . 8 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { B , A } ) )
89	88	impcom 430	. . . . . . 7 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> -. ( <. V , E >. Neighbors A ) = { B , A } )
90	89	3adant3 1016	. . . . . 6 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> -. ( <. V , E >. Neighbors A ) = { B , A } )
91		biorf 405	. . . . . . 7 |- ( -. ( <. V , E >. Neighbors A ) = { B , A } -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( ( <. V , E >. Neighbors A ) = { B , A } \/ ( <. V , E >. Neighbors A ) = { B , C } ) ) )
92		orcom 387	. . . . . . 7 |- ( ( ( <. V , E >. Neighbors A ) = { B , A } \/ ( <. V , E >. Neighbors A ) = { B , C } ) <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) )
93	91, 92	syl6bb 261	. . . . . 6 |- ( -. ( <. V , E >. Neighbors A ) = { B , A } -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) ) )
94	90, 93	syl 16	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) ) )
95		prid2g 4139	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { C , A } )
96	95	3ad2ant1 1017	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { C , A } )
97		eleq2 2530	. . . . . . . . . . . . 13 |- ( ( <. V , E >. Neighbors A ) = { C , A } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { C , A } ) )
98	96, 97	syl5ibrcom 222	. . . . . . . . . . . 12 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( <. V , E >. Neighbors A ) = { C , A } -> A e. ( <. V , E >. Neighbors A ) ) )
99	98	con3rr3 136	. . . . . . . . . . 11 |- ( -. A e. ( <. V , E >. Neighbors A ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { C , A } ) )
100	80, 99	sylbi 195	. . . . . . . . . 10 |- ( A e/ ( <. V , E >. Neighbors A ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { C , A } ) )
101	79, 100	syl 16	. . . . . . . . 9 |- ( V USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { C , A } ) )
102	101	adantl 466	. . . . . . . 8 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { C , A } ) )
103	102	impcom 430	. . . . . . 7 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> -. ( <. V , E >. Neighbors A ) = { C , A } )
104	103	3adant3 1016	. . . . . 6 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> -. ( <. V , E >. Neighbors A ) = { C , A } )
105		biorf 405	. . . . . 6 |- ( -. ( <. V , E >. Neighbors A ) = { C , A } -> ( ( <. V , E >. Neighbors A ) = { C , B } <-> ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) )
106	104, 105	syl 16	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( <. V , E >. Neighbors A ) = { C , B } <-> ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) )
107	94, 106	orbi12d 709	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { C , B } ) <-> ( ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) \/ ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) ) )
108		prid1g 4138	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , B } )
109	108	3ad2ant1 1017	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , B } )
110		eleq2 2530	. . . . . . . . . . . . . . 15 |- ( ( <. V , E >. Neighbors A ) = { A , B } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { A , B } ) )
111	109, 110	syl5ibrcom 222	. . . . . . . . . . . . . 14 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( <. V , E >. Neighbors A ) = { A , B } -> A e. ( <. V , E >. Neighbors A ) ) )
112	111	con3dimp 441	. . . . . . . . . . . . 13 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ -. A e. ( <. V , E >. Neighbors A ) ) -> -. ( <. V , E >. Neighbors A ) = { A , B } )
113		prid1g 4138	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , C } )
114	113	3ad2ant1 1017	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , C } )
115		eleq2 2530	. . . . . . . . . . . . . . 15 |- ( ( <. V , E >. Neighbors A ) = { A , C } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { A , C } ) )
116	114, 115	syl5ibrcom 222	. . . . . . . . . . . . . 14 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( ( <. V , E >. Neighbors A ) = { A , C } -> A e. ( <. V , E >. Neighbors A ) ) )
117	116	con3dimp 441	. . . . . . . . . . . . 13 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ -. A e. ( <. V , E >. Neighbors A ) ) -> -. ( <. V , E >. Neighbors A ) = { A , C } )
118	112, 117	jca 532	. . . . . . . . . . . 12 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ -. A e. ( <. V , E >. Neighbors A ) ) -> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) )
119	118	expcom 435	. . . . . . . . . . 11 |- ( -. A e. ( <. V , E >. Neighbors A ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) ) )
120	80, 119	sylbi 195	. . . . . . . . . 10 |- ( A e/ ( <. V , E >. Neighbors A ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) ) )
121	79, 120	syl 16	. . . . . . . . 9 |- ( V USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) ) )
122	121	adantl 466	. . . . . . . 8 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) ) )
123	122	impcom 430	. . . . . . 7 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) ) -> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) )
124	123	3adant3 1016	. . . . . 6 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) )
125		ioran 490	. . . . . 6 |- ( -. ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) <-> ( -. ( <. V , E >. Neighbors A ) = { A , B } /\ -. ( <. V , E >. Neighbors A ) = { A , C } ) )
126	124, 125	sylibr 212	. . . . 5 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> -. ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) )
127	126	3bior1fd 1334	. . . 4 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) \/ ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) <-> ( ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) \/ ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) \/ ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) ) )
128	78, 107, 127	3bitrd 279	. . 3 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( ( ( <. V , E >. Neighbors A ) = { A , B } \/ ( <. V , E >. Neighbors A ) = { A , C } ) \/ ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , A } ) \/ ( ( <. V , E >. Neighbors A ) = { C , A } \/ ( <. V , E >. Neighbors A ) = { C , B } ) ) ) )
129	44, 72, 128	3bitr4rd 286	. 2 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> ( E. w e. ( { A , B , C } \ { A } ) ( <. V , E >. Neighbors A ) = { A , w } \/ E. w e. ( { A , B , C } \ { B } ) ( <. V , E >. Neighbors A ) = { B , w } \/ E. w e. ( { A , B , C } \ { C } ) ( <. V , E >. Neighbors A ) = { C , w } ) ) )
130	17, 23, 129	3bitr4rd 286	1 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ ( V = { A , B , C } /\ V USGrph E ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> ( ( <. V , E >. Neighbors A ) = { B , C } <-> E. v e. V E. w e. ( V \ { v } ) ( <. V , E >. Neighbors A ) = { v , w } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/ w3o 972   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   e/ wnel 2653   E.wrex 2808   \ cdif 3468   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456  (class class class)co 6296   USGrph cusg 24457   Neighbors cnbgra 24544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460  df-nbgra 24547

This theorem is referenced by:  nb3grapr  24580