Theorem nb3graprlem2 25192

Description: Lemma 2 for nb3grapr 25193. (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 3980	. . . . . 6 |- ( v = A -> { v } = { A } )
2	1	difeq2d 3553	. . . . 5 |- ( v = A -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { A } ) )
3		preq1 4054	. . . . . 6 |- ( v = A -> { v , w } = { A , w } )
4	3	eqeq2d 2463	. . . . 5 |- ( v = A -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { A , w } ) )
5	2, 4	rexeqbidv 3004	. . . 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 3980	. . . . . 6 |- ( v = B -> { v } = { B } )
7	6	difeq2d 3553	. . . . 5 |- ( v = B -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { B } ) )
8		preq1 4054	. . . . . 6 |- ( v = B -> { v , w } = { B , w } )
9	8	eqeq2d 2463	. . . . 5 |- ( v = B -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { B , w } ) )
10	7, 9	rexeqbidv 3004	. . . 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 3980	. . . . . 6 |- ( v = C -> { v } = { C } )
12	11	difeq2d 3553	. . . . 5 |- ( v = C -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { C } ) )
13		preq1 4054	. . . . . 6 |- ( v = C -> { v , w } = { C , w } )
14	13	eqeq2d 2463	. . . . 5 |- ( v = C -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { C , w } ) )
15	12, 14	rexeqbidv 3004	. . . 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 4026	. . 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 1030	. 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 459	. . . 4 |- ( ( V = { A , B , C } /\ V USGrph E ) -> V = { A , B , C } )
19		difeq1 3546	. . . . . 6 |- ( V = { A , B , C } -> ( V \ { v } ) = ( { A , B , C } \ { v } ) )
20	19	adantr 467	. . . . 5 |- ( ( V = { A , B , C } /\ V USGrph E ) -> ( V \ { v } ) = ( { A , B , C } \ { v } ) )
21	20	rexeqdv 2996	. . . 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 3004	. . 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 1031	. 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 4055	. . . . . . . 8 |- ( w = B -> { A , w } = { A , B } )
25	24	eqeq2d 2463	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , B } ) )
26		preq2 4055	. . . . . . . 8 |- ( w = C -> { A , w } = { A , C } )
27	26	eqeq2d 2463	. . . . . . 7 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , C } ) )
28	25, 27	rexprg 4024	. . . . . 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 1027	. . . . 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 4055	. . . . . . . . 9 |- ( w = C -> { B , w } = { B , C } )
31	30	eqeq2d 2463	. . . . . . . 8 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , C } ) )
32		preq2 4055	. . . . . . . . 9 |- ( w = A -> { B , w } = { B , A } )
33	32	eqeq2d 2463	. . . . . . . 8 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , A } ) )
34	31, 33	rexprg 4024	. . . . . . 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 455	. . . . . 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 1028	. . . . 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 4055	. . . . . . . 8 |- ( w = A -> { C , w } = { C , A } )
38	37	eqeq2d 2463	. . . . . . 7 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , A } ) )
39		preq2 4055	. . . . . . . 8 |- ( w = B -> { C , w } = { C , B } )
40	39	eqeq2d 2463	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , B } ) )
41	38, 40	rexprg 4024	. . . . . 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 1029	. . . . 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 1340	. . . 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 1030	. . 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 4070	. . . . . . . . 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 3552	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = ( { B , C , A } \ { A } ) )
48		necom 2679	. . . . . . . . 9 |- ( A =/= B <-> B =/= A )
49		necom 2679	. . . . . . . . 9 |- ( A =/= C <-> C =/= A )
50		diftpsn3 4113	. . . . . . . . 9 |- ( ( B =/= A /\ C =/= A ) -> ( { B , C , A } \ { A } ) = { B , C } )
51	48, 49, 50	syl2anb 482	. . . . . . . 8 |- ( ( A =/= B /\ A =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
52	51	3adant3 1029	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
53	47, 52	eqtrd 2487	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = { B , C } )
54	53	rexeqdv 2996	. . . . 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 4070	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
56	55	eqcomi 2462	. . . . . . . . 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 3552	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = ( { C , A , B } \ { B } ) )
59		necom 2679	. . . . . . . . . . . 12 |- ( B =/= C <-> C =/= B )
60	59	anbi1i 702	. . . . . . . . . . 11 |- ( ( B =/= C /\ A =/= B ) <-> ( C =/= B /\ A =/= B ) )
61	60	biimpi 198	. . . . . . . . . 10 |- ( ( B =/= C /\ A =/= B ) -> ( C =/= B /\ A =/= B ) )
62	61	ancoms 455	. . . . . . . . 9 |- ( ( A =/= B /\ B =/= C ) -> ( C =/= B /\ A =/= B ) )
63		diftpsn3 4113	. . . . . . . . 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 1028	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
66	58, 65	eqtrd 2487	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = { C , A } )
67	66	rexeqdv 2996	. . . . 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 4113	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
69	68	3adant1 1027	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
70	69	rexeqdv 2996	. . . . 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 1340	. . . 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 1032	. . 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 4053	. . . . . . . 8 |- { C , B } = { B , C }
74	73	eqeq2i 2465	. . . . . . 7 |- ( ( <. V , E >. Neighbors A ) = { C , B } <-> ( <. V , E >. Neighbors A ) = { B , C } )
75	74	orbi2i 522	. . . . . 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 517	. . . . . 6 |- ( ( ( <. V , E >. Neighbors A ) = { B , C } \/ ( <. V , E >. Neighbors A ) = { B , C } ) <-> ( <. V , E >. Neighbors A ) = { B , C } )
77	75, 76	bitr2i 254	. . . . 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 25176	. . . . . . . . . 10 |- ( V USGrph E -> A e/ ( <. V , E >. Neighbors A ) )
80		df-nel 2627	. . . . . . . . . . 11 |- ( A e/ ( <. V , E >. Neighbors A ) <-> -. A e. ( <. V , E >. Neighbors A ) )
81		prid2g 4082	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { B , A } )
82	81	3ad2ant1 1030	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { B , A } )
83		eleq2 2520	. . . . . . . . . . . . 13 |- ( ( <. V , E >. Neighbors A ) = { B , A } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { B , A } ) )
84	82, 83	syl5ibrcom 226	. . . . . . . . . . . 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 142	. . . . . . . . . . 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 199	. . . . . . . . . 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 468	. . . . . . . 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 432	. . . . . . 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 1029	. . . . . 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 407	. . . . . . 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 389	. . . . . . 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 265	. . . . . 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 4082	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { C , A } )
96	95	3ad2ant1 1030	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { C , A } )
97		eleq2 2520	. . . . . . . . . . . . 13 |- ( ( <. V , E >. Neighbors A ) = { C , A } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { C , A } ) )
98	96, 97	syl5ibrcom 226	. . . . . . . . . . . 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 142	. . . . . . . . . . 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 199	. . . . . . . . . 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 468	. . . . . . . 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 432	. . . . . . 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 1029	. . . . . 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 407	. . . . . 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 717	. . . 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 4081	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , B } )
109	108	3ad2ant1 1030	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , B } )
110		eleq2 2520	. . . . . . . . . . . . . . 15 |- ( ( <. V , E >. Neighbors A ) = { A , B } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { A , B } ) )
111	109, 110	syl5ibrcom 226	. . . . . . . . . . . . . 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 443	. . . . . . . . . . . . 13 |- ( ( ( A e. X /\ B e. Y /\ C e. Z ) /\ -. A e. ( <. V , E >. Neighbors A ) ) -> -. ( <. V , E >. Neighbors A ) = { A , B } )
113		prid1g 4081	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , C } )
114	113	3ad2ant1 1030	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , C } )
115		eleq2 2520	. . . . . . . . . . . . . . 15 |- ( ( <. V , E >. Neighbors A ) = { A , C } -> ( A e. ( <. V , E >. Neighbors A ) <-> A e. { A , C } ) )
116	114, 115	syl5ibrcom 226	. . . . . . . . . . . . . 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 443	. . . . . . . . . . . . 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 535	. . . . . . . . . . . 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 437	. . . . . . . . . . 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 199	. . . . . . . . . 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 468	. . . . . . . 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 432	. . . . . . 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 1029	. . . . . 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 493	. . . . . 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 216	. . . . 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 1377	. . . 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 283	. . 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 290	. 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 290	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 188   \/ wo 370   /\ wa 371   \/ w3o 985   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   e/ wnel 2625   E.wrex 2740   \ cdif 3403   {csn 3970   {cpr 3972   {ctp 3974   <.cop 3976   class class class wbr 4405  (class class class)co 6295   USGrph cusg 25069   Neighbors cnbgra 25157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523  df-usgra 25072  df-nbgra 25160

This theorem is referenced by:  nb3grapr  25193