Theorem nb3graprlem2 25259

Description: Lemma 2 for nb3grapr 25260. (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 3969	. . . . . 6 |- ( v = A -> { v } = { A } )
2	1	difeq2d 3540	. . . . 5 |- ( v = A -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { A } ) )
3		preq1 4042	. . . . . 6 |- ( v = A -> { v , w } = { A , w } )
4	3	eqeq2d 2481	. . . . 5 |- ( v = A -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { A , w } ) )
5	2, 4	rexeqbidv 2988	. . . 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 3969	. . . . . 6 |- ( v = B -> { v } = { B } )
7	6	difeq2d 3540	. . . . 5 |- ( v = B -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { B } ) )
8		preq1 4042	. . . . . 6 |- ( v = B -> { v , w } = { B , w } )
9	8	eqeq2d 2481	. . . . 5 |- ( v = B -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { B , w } ) )
10	7, 9	rexeqbidv 2988	. . . 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 3969	. . . . . 6 |- ( v = C -> { v } = { C } )
12	11	difeq2d 3540	. . . . 5 |- ( v = C -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { C } ) )
13		preq1 4042	. . . . . 6 |- ( v = C -> { v , w } = { C , w } )
14	13	eqeq2d 2481	. . . . 5 |- ( v = C -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { C , w } ) )
15	12, 14	rexeqbidv 2988	. . . 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 4015	. . 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 1051	. 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 464	. . . 4 |- ( ( V = { A , B , C } /\ V USGrph E ) -> V = { A , B , C } )
19		difeq1 3533	. . . . . 6 |- ( V = { A , B , C } -> ( V \ { v } ) = ( { A , B , C } \ { v } ) )
20	19	adantr 472	. . . . 5 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V \ { v } ) = ( { A , B , C } \ { v } ) )
21	20	rexeqdv 2980	. . . 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 2988	. . 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 1052	. 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 4043	. . . . . . . 8 |- ( w = B -> { A , w } = { A , B } )
25	24	eqeq2d 2481	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , B } ) )
26		preq2 4043	. . . . . . . 8 |- ( w = C -> { A , w } = { A , C } )
27	26	eqeq2d 2481	. . . . . . 7 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , C } ) )
28	25, 27	rexprg 4013	. . . . . 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 1048	. . . . 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 4043	. . . . . . . . 9 |- ( w = C -> { B , w } = { B , C } )
31	30	eqeq2d 2481	. . . . . . . 8 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , C } ) )
32		preq2 4043	. . . . . . . . 9 |- ( w = A -> { B , w } = { B , A } )
33	32	eqeq2d 2481	. . . . . . . 8 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , A } ) )
34	31, 33	rexprg 4013	. . . . . . 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 460	. . . . . 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 1049	. . . . 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 4043	. . . . . . . 8 |- ( w = A -> { C , w } = { C , A } )
38	37	eqeq2d 2481	. . . . . . 7 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , A } ) )
39		preq2 4043	. . . . . . . 8 |- ( w = B -> { C , w } = { C , B } )
40	39	eqeq2d 2481	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , B } ) )
41	38, 40	rexprg 4013	. . . . . 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 1050	. . . . 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 1364	. . . 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 1051	. . 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 4058	. . . . . . . . 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 3539	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = ( { B , C , A } \ { A } ) )
48		necom 2696	. . . . . . . . 9 |- ( A =/= B <-> B =/= A )
49		necom 2696	. . . . . . . . 9 |- ( A =/= C <-> C =/= A )
50		diftpsn3 4101	. . . . . . . . 9 |- ( ( B =/= A /\ C =/= A ) -> ( { B , C , A } \ { A } ) = { B , C } )
51	48, 49, 50	syl2anb 487	. . . . . . . 8 |- ( ( A =/= B /\ A =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
52	51	3adant3 1050	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
53	47, 52	eqtrd 2505	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = { B , C } )
54	53	rexeqdv 2980	. . . . 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 4058	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
56	55	eqcomi 2480	. . . . . . . . 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 3539	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = ( { C , A , B } \ { B } ) )
59		necom 2696	. . . . . . . . . . . 12 |- ( B =/= C <-> C =/= B )
60	59	anbi1i 709	. . . . . . . . . . 11 |- ( ( B =/= C /\ A =/= B ) <-> ( C =/= B /\ A =/= B ) )
61	60	biimpi 199	. . . . . . . . . 10 |- ( ( B =/= C /\ A =/= B ) -> ( C =/= B /\ A =/= B ) )
62	61	ancoms 460	. . . . . . . . 9 |- ( ( A =/= B /\ B =/= C ) -> ( C =/= B /\ A =/= B ) )
63		diftpsn3 4101	. . . . . . . . 9 |- ( ( C =/= B /\ A =/= B ) -> ( { C , A , B } \ { B } ) = { C , A } )
64	62, 63	syl 17	. . . . . . . 8 |- ( ( A =/= B /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
65	64	3adant2 1049	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
66	58, 65	eqtrd 2505	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = { C , A } )
67	66	rexeqdv 2980	. . . . 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 4101	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
69	68	3adant1 1048	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
70	69	rexeqdv 2980	. . . . 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 1364	. . . 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 1053	. . 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 4041	. . . . . . . 8 |- { C , B } = { B , C }
74	73	eqeq2i 2483	. . . . . . 7 |- ( ( <. V , E >. Neighbors A ) = { C , B } <-> ( <. V , E >. Neighbors A ) = { B , C } )
75	74	orbi2i 528	. . . . . 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 523	. . . . . 6 |- ( ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , C } ) <-> ( <. V , E >. Neighbors A ) = { B , C } )
77	75, 76	bitr2i 258	. . . . 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 25243	. . . . . . . . . 10 |- ( V USGrph E -> A e/ ( <. V , E >. Neighbors A ) )
80		df-nel 2644	. . . . . . . . . . 11 |- ( A e/ ( <. V , E >. Neighbors A ) <-> -. A e. ( <. V , E >. Neighbors A ) )
81		prid2g 4070	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { B , A } )
82	81	3ad2ant1 1051	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { B , A } )
83		eleq2 2538	. . . . . . . . . . . . 13 |- ( ( <. V , E >. Neighbors A ) = { B , A } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { B , A } ) )
84	82, 83	syl5ibrcom 230	. . . . . . . . . . . 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 143	. . . . . . . . . . 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 200	. . . . . . . . . 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 17	. . . . . . . . 9 |- ( V USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { B , A } ) )
88	87	adantl 473	. . . . . . . 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 437	. . . . . . 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 1050	. . . . . 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 412	. . . . . . 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 394	. . . . . . 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 269	. . . . . 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 17	. . . . 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 4070	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { C , A } )
96	95	3ad2ant1 1051	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { C , A } )
97		eleq2 2538	. . . . . . . . . . . . 13 |- ( ( <. V , E >. Neighbors A ) = { C , A } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { C , A } ) )
98	96, 97	syl5ibrcom 230	. . . . . . . . . . . 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 143	. . . . . . . . . . 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 200	. . . . . . . . . 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 17	. . . . . . . . 9 |- ( V USGrph E -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> -. ( <. V , E >. Neighbors A ) = { C , A } ) )
102	101	adantl 473	. . . . . . . 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 437	. . . . . . 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 1050	. . . . . 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 412	. . . . . 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 17	. . . . 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 724	. . . 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 4069	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , B } )
109	108	3ad2ant1 1051	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , B } )
110		eleq2 2538	. . . . . . . . . . . . . . 15 |- ( ( <. V , E >. Neighbors A ) = { A , B } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { A , B } ) )
111	109, 110	syl5ibrcom 230	. . . . . . . . . . . . . 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 448	. . . . . . . . . . . . 13 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ -. A e. ( <. V , E >. Neighbors A ) ) -> -. ( <. V , E >. Neighbors A ) = { A , B } )
113		prid1g 4069	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , C } )
114	113	3ad2ant1 1051	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , C } )
115		eleq2 2538	. . . . . . . . . . . . . . 15 |- ( ( <. V , E >. Neighbors A ) = { A , C } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { A , C } ) )
116	114, 115	syl5ibrcom 230	. . . . . . . . . . . . . 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 448	. . . . . . . . . . . . 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 541	. . . . . . . . . . . 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 442	. . . . . . . . . . 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 200	. . . . . . . . . 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 17	. . . . . . . . 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 473	. . . . . . . 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 437	. . . . . . 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 1050	. . . . . 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 498	. . . . . 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 217	. . . . 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 1403	. . . 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 287	. . 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 294	. 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 294	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 189   \/ wo 375   /\ wa 376   \/ w3o 1006   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   e/ wnel 2642   E.wrex 2757   \ cdif 3387   {csn 3959   {cpr 3961   {ctp 3963   <.cop 3965   class class class wbr 4395  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227

This theorem is referenced by:  nb3grapr  25260