Theorem nb3graprlem2 23328

Description: Lemma 2 for nb3grapr 23329. (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 3882	. . . . . 6 |- ( v = A -> { v } = { A } )
2	1	difeq2d 3469	. . . . 5 |- ( v = A -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { A } ) )
3		preq1 3949	. . . . . 6 |- ( v = A -> { v , w } = { A , w } )
4	3	eqeq2d 2449	. . . . 5 |- ( v = A -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { A , w } ) )
5	2, 4	rexeqbidv 2927	. . . 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 3882	. . . . . 6 |- ( v = B -> { v } = { B } )
7	6	difeq2d 3469	. . . . 5 |- ( v = B -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { B } ) )
8		preq1 3949	. . . . . 6 |- ( v = B -> { v , w } = { B , w } )
9	8	eqeq2d 2449	. . . . 5 |- ( v = B -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { B , w } ) )
10	7, 9	rexeqbidv 2927	. . . 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 3882	. . . . . 6 |- ( v = C -> { v } = { C } )
12	11	difeq2d 3469	. . . . 5 |- ( v = C -> ( { A , B , C } \ { v } ) = ( { A , B , C } \ { C } ) )
13		preq1 3949	. . . . . 6 |- ( v = C -> { v , w } = { C , w } )
14	13	eqeq2d 2449	. . . . 5 |- ( v = C -> ( ( <. V , E >. Neighbors A ) = { v , w } <-> ( <. V , E >. Neighbors A ) = { C , w } ) )
15	12, 14	rexeqbidv 2927	. . . 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 3923	. . 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 1009	. 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 3462	. . . . . 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 2919	. . . 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 2927	. . 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 1010	. 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 3950	. . . . . . . 8 |- ( w = B -> { A , w } = { A , B } )
25	24	eqeq2d 2449	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , B } ) )
26		preq2 3950	. . . . . . . 8 |- ( w = C -> { A , w } = { A , C } )
27	26	eqeq2d 2449	. . . . . . 7 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { A , w } <-> ( <. V , E >. Neighbors A ) = { A , C } ) )
28	25, 27	rexprg 3921	. . . . . 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 1006	. . . . 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 3950	. . . . . . . . 9 |- ( w = C -> { B , w } = { B , C } )
31	30	eqeq2d 2449	. . . . . . . 8 |- ( w = C -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , C } ) )
32		preq2 3950	. . . . . . . . 9 |- ( w = A -> { B , w } = { B , A } )
33	32	eqeq2d 2449	. . . . . . . 8 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { B , w } <-> ( <. V , E >. Neighbors A ) = { B , A } ) )
34	31, 33	rexprg 3921	. . . . . . 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 1007	. . . . 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 3950	. . . . . . . 8 |- ( w = A -> { C , w } = { C , A } )
38	37	eqeq2d 2449	. . . . . . 7 |- ( w = A -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , A } ) )
39		preq2 3950	. . . . . . . 8 |- ( w = B -> { C , w } = { C , B } )
40	39	eqeq2d 2449	. . . . . . 7 |- ( w = B -> ( ( <. V , E >. Neighbors A ) = { C , w } <-> ( <. V , E >. Neighbors A ) = { C , B } ) )
41	38, 40	rexprg 3921	. . . . . 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 1008	. . . . 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 1288	. . . 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 1009	. . 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 3965	. . . . . . . . 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 3468	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = ( { B , C , A } \ { A } ) )
48		necom 2688	. . . . . . . . 9 |- ( A =/= B <-> B =/= A )
49		necom 2688	. . . . . . . . 9 |- ( A =/= C <-> C =/= A )
50		diftpsn3 4007	. . . . . . . . 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 1008	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { B , C , A } \ { A } ) = { B , C } )
53	47, 52	eqtrd 2470	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { A } ) = { B , C } )
54	53	rexeqdv 2919	. . . . 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 3965	. . . . . . . . . 10 |- { C , A , B } = { A , B , C }
56	55	eqcomi 2442	. . . . . . . . 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 3468	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = ( { C , A , B } \ { B } ) )
59		necom 2688	. . . . . . . . . . . 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 4007	. . . . . . . . 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 1007	. . . . . . 7 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { C , A , B } \ { B } ) = { C , A } )
66	58, 65	eqtrd 2470	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { B } ) = { C , A } )
67	66	rexeqdv 2919	. . . . 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 4007	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
69	68	3adant1 1006	. . . . . 6 |- ( ( A =/= B /\ A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )
70	69	rexeqdv 2919	. . . . 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 1288	. . . 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 1011	. . 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 3948	. . . . . . . 8 |- { C , B } = { B , C }
74	73	eqeq2i 2448	. . . . . . 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 23315	. . . . . . . . . 10 |- ( V USGrph E -> A e/ ( <. V , E >. Neighbors A ) )
80		df-nel 2604	. . . . . . . . . . 11 |- ( A e/ ( <. V , E >. Neighbors A ) <-> -. A e. ( <. V , E >. Neighbors A ) )
81		prid2g 3977	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { B , A } )
82	81	3ad2ant1 1009	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { B , A } )
83		eleq2 2499	. . . . . . . . . . . . 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 1008	. . . . . 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 3977	. . . . . . . . . . . . . 14 |- ( A e. X -> A e. { C , A } )
96	95	3ad2ant1 1009	. . . . . . . . . . . . 13 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { C , A } )
97		eleq2 2499	. . . . . . . . . . . . 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 1008	. . . . . 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 3976	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , B } )
109	108	3ad2ant1 1009	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , B } )
110		eleq2 2499	. . . . . . . . . . . . . . 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 3976	. . . . . . . . . . . . . . . 16 |- ( A e. X -> A e. { A , C } )
114	113	3ad2ant1 1009	. . . . . . . . . . . . . . 15 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> A e. { A , C } )
115		eleq2 2499	. . . . . . . . . . . . . . 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 1008	. . . . . 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 1324	. . . 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 964   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2601   e/ wnel 2602   E.wrex 2711   \ cdif 3320   {csn 3872   {cpr 3874   {ctp 3876   <.cop 3878   class class class wbr 4287  (class class class)co 6086   USGrph cusg 23232   Neighbors cnbgra 23297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096  df-usgra 23234  df-nbgra 23300

This theorem is referenced by:  nb3grapr  23329