Theorem frgrreg 27252

Description: If a finite nonempty friendship graph is K-regular, then K must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 3-Jun-2021.)

Hypothesis

Ref	Expression
frgrreggt1.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
frgrreg	|- ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( K = 0 \/ K = 2 ) ) )

Proof of Theorem frgrreg

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ancom 466	. . . . . . . . 9 |- ( ( V e. Fin /\ V =/= (/) ) <-> ( V =/= (/) /\ V e. Fin ) )
2		ancom 466	. . . . . . . . 9 |- ( ( G e. FriendGraph /\ G RegUSGraph K ) <-> ( G RegUSGraph K /\ G e. FriendGraph ) )
3	1, 2	anbi12i 733	. . . . . . . 8 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) <-> ( ( V =/= (/) /\ V e. Fin ) /\ ( G RegUSGraph K /\ G e. FriendGraph ) ) )
4	3	biimpi 206	. . . . . . 7 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( ( V =/= (/) /\ V e. Fin ) /\ ( G RegUSGraph K /\ G e. FriendGraph ) ) )
5	4	ancomd 467	. . . . . 6 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) )
6		frgrreggt1.v	. . . . . . 7 |- V = (Vtx ` G )
7	6	numclwwlk7lem 27247	. . . . . 6 |- ( ( ( G RegUSGraph K /\ G e. FriendGraph ) /\ ( V =/= (/) /\ V e. Fin ) ) -> K e. NN0 )
8	5, 7	syl 17	. . . . 5 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> K e. NN0 )
9		neanior 2886	. . . . . . . 8 |- ( ( K =/= 0 /\ K =/= 2 ) <-> -. ( K = 0 \/ K = 2 ) )
10		nn0re 11301	. . . . . . . . . . . . . 14 |- ( K e. NN0 -> K e. RR )
11		1re 10039	. . . . . . . . . . . . . 14 |- 1 e. RR
12		lenlt 10116	. . . . . . . . . . . . . 14 |- ( ( K e. RR /\ 1 e. RR ) -> ( K <_ 1 <-> -. 1 < K ) )
13	10, 11, 12	sylancl 694	. . . . . . . . . . . . 13 |- ( K e. NN0 -> ( K <_ 1 <-> -. 1 < K ) )
14	13	adantl 482	. . . . . . . . . . . 12 |- ( ( ( K =/= 0 /\ K =/= 2 ) /\ K e. NN0 ) -> ( K <_ 1 <-> -. 1 < K ) )
15		elnnne0 11306	. . . . . . . . . . . . . . . 16 |- ( K e. NN <-> ( K e. NN0 /\ K =/= 0 ) )
16		nnle1eq1 11048	. . . . . . . . . . . . . . . . 17 |- ( K e. NN -> ( K <_ 1 <-> K = 1 ) )
17	16	biimpd 219	. . . . . . . . . . . . . . . 16 |- ( K e. NN -> ( K <_ 1 -> K = 1 ) )
18	15, 17	sylbir 225	. . . . . . . . . . . . . . 15 |- ( ( K e. NN0 /\ K =/= 0 ) -> ( K <_ 1 -> K = 1 ) )
19	18	a1d 25	. . . . . . . . . . . . . 14 |- ( ( K e. NN0 /\ K =/= 0 ) -> ( K =/= 2 -> ( K <_ 1 -> K = 1 ) ) )
20	19	expimpd 629	. . . . . . . . . . . . 13 |- ( K e. NN0 -> ( ( K =/= 0 /\ K =/= 2 ) -> ( K <_ 1 -> K = 1 ) ) )
21	20	impcom 446	. . . . . . . . . . . 12 |- ( ( ( K =/= 0 /\ K =/= 2 ) /\ K e. NN0 ) -> ( K <_ 1 -> K = 1 ) )
22	14, 21	sylbird 250	. . . . . . . . . . 11 |- ( ( ( K =/= 0 /\ K =/= 2 ) /\ K e. NN0 ) -> ( -. 1 < K -> K = 1 ) )
23	6	fveq2i 6194	. . . . . . . . . . . . . . . . . 18 |- ( # ` V ) = ( # ` (Vtx ` G ) )
24	23	eqeq1i 2627	. . . . . . . . . . . . . . . . 17 |- ( ( # ` V ) = 1 <-> ( # ` (Vtx ` G ) ) = 1 )
25	24	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( ( # ` V ) = 1 -> ( # ` (Vtx ` G ) ) = 1 )
26		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( G e. FriendGraph /\ G RegUSGraph K ) -> G RegUSGraph K )
27	26	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> G RegUSGraph K )
28		rusgr1vtx 26484	. . . . . . . . . . . . . . . 16 |- ( ( ( # ` (Vtx ` G ) ) = 1 /\ G RegUSGraph K ) -> K = 0 )
29	25, 27, 28	syl2an 494	. . . . . . . . . . . . . . 15 |- ( ( ( # ` V ) = 1 /\ ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) ) -> K = 0 )
30	29	orcd 407	. . . . . . . . . . . . . 14 |- ( ( ( # ` V ) = 1 /\ ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) ) -> ( K = 0 \/ K = 2 ) )
31	30	ex 450	. . . . . . . . . . . . 13 |- ( ( # ` V ) = 1 -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) )
32	31	a1d 25	. . . . . . . . . . . 12 |- ( ( # ` V ) = 1 -> ( K = 1 -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) ) )
33		eqid 2622	. . . . . . . . . . . . . . . . 17 |- (VtxDeg ` G ) = (VtxDeg ` G )
34	6, 33	rusgrprop0 26463	. . . . . . . . . . . . . . . 16 |- ( G RegUSGraph K -> ( G e. USGraph /\ K e. NN0* /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) )
35		simp2 1062	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -. ( # ` V ) = 1 /\ G e. FriendGraph /\ ( V e. Fin /\ V =/= (/) ) ) -> G e. FriendGraph )
36		hashnncl 13157	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( V e. Fin -> ( ( # ` V ) e. NN <-> V =/= (/) ) )
37		df-ne 2795	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` V ) =/= 1 <-> -. ( # ` V ) = 1 )
38		nngt1ne1 11047	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` V ) e. NN -> ( 1 < ( # ` V ) <-> ( # ` V ) =/= 1 ) )
39	38	biimprd 238	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` V ) e. NN -> ( ( # ` V ) =/= 1 -> 1 < ( # ` V ) ) )
40	37, 39	syl5bir 233	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` V ) e. NN -> ( -. ( # ` V ) = 1 -> 1 < ( # ` V ) ) )
41	36, 40	syl6bir 244	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( V e. Fin -> ( V =/= (/) -> ( -. ( # ` V ) = 1 -> 1 < ( # ` V ) ) ) )
42	41	imp 445	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( V e. Fin /\ V =/= (/) ) -> ( -. ( # ` V ) = 1 -> 1 < ( # ` V ) ) )
43	42	impcom 446	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( -. ( # ` V ) = 1 /\ ( V e. Fin /\ V =/= (/) ) ) -> 1 < ( # ` V ) )
44	6	vdgn1frgrv3 27161	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) -> A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 )
45	35, 43, 44	3imp3i2an 1278	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. ( # ` V ) = 1 /\ G e. FriendGraph /\ ( V e. Fin /\ V =/= (/) ) ) -> A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 )
46		r19.26 3064	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( A. v e. V ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) <-> ( A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) )
47		r19.2z 4060	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( V =/= (/) /\ A. v e. V ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) ) -> E. v e. V ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) )
48		neeq1 2856	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( (VtxDeg ` G ) ` v ) = K -> ( ( (VtxDeg ` G ) ` v ) =/= 1 <-> K =/= 1 ) )
49	48	biimpd 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( (VtxDeg ` G ) ` v ) = K -> ( ( (VtxDeg ` G ) ` v ) =/= 1 -> K =/= 1 ) )
50	49	impcom 446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) -> K =/= 1 )
51		eqneqall 2805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( K = 1 -> ( K =/= 1 -> ( K = 0 \/ K = 2 ) ) )
52	51	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( K =/= 1 -> ( K = 1 -> ( K = 0 \/ K = 2 ) ) )
53	50, 52	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( K = 1 -> ( K = 0 \/ K = 2 ) ) )
54	53	rexlimivw 3029	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( E. v e. V ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( K = 1 -> ( K = 0 \/ K = 2 ) ) )
55	47, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( V =/= (/) /\ A. v e. V ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) ) -> ( K = 1 -> ( K = 0 \/ K = 2 ) ) )
56	55	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( V =/= (/) -> ( A. v e. V ( ( (VtxDeg ` G ) ` v ) =/= 1 /\ ( (VtxDeg ` G ) ` v ) = K ) -> ( K = 1 -> ( K = 0 \/ K = 2 ) ) ) )
57	46, 56	syl5bir 233	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( V =/= (/) -> ( ( A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> ( K = 1 -> ( K = 0 \/ K = 2 ) ) ) )
58	57	expd 452	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( V =/= (/) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( K = 1 -> ( K = 0 \/ K = 2 ) ) ) ) )
59	58	com34 91	. . . . . . . . . . . . . . . . . . . . . 22 |- ( V =/= (/) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 -> ( K = 1 -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( K = 0 \/ K = 2 ) ) ) ) )
60	59	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( V e. Fin /\ V =/= (/) ) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 -> ( K = 1 -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( K = 0 \/ K = 2 ) ) ) ) )
61	60	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. ( # ` V ) = 1 /\ G e. FriendGraph /\ ( V e. Fin /\ V =/= (/) ) ) -> ( A. v e. V ( (VtxDeg ` G ) ` v ) =/= 1 -> ( K = 1 -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( K = 0 \/ K = 2 ) ) ) ) )
62	45, 61	mpd 15	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. ( # ` V ) = 1 /\ G e. FriendGraph /\ ( V e. Fin /\ V =/= (/) ) ) -> ( K = 1 -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( K = 0 \/ K = 2 ) ) ) )
63	62	3exp 1264	. . . . . . . . . . . . . . . . . 18 |- ( -. ( # ` V ) = 1 -> ( G e. FriendGraph -> ( ( V e. Fin /\ V =/= (/) ) -> ( K = 1 -> ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( K = 0 \/ K = 2 ) ) ) ) ) )
64	63	com15 101	. . . . . . . . . . . . . . . . 17 |- ( A. v e. V ( (VtxDeg ` G ) ` v ) = K -> ( G e. FriendGraph -> ( ( V e. Fin /\ V =/= (/) ) -> ( K = 1 -> ( -. ( # ` V ) = 1 -> ( K = 0 \/ K = 2 ) ) ) ) ) )
65	64	3ad2ant3 1084	. . . . . . . . . . . . . . . 16 |- ( ( G e. USGraph /\ K e. NN0* /\ A. v e. V ( (VtxDeg ` G ) ` v ) = K ) -> ( G e. FriendGraph -> ( ( V e. Fin /\ V =/= (/) ) -> ( K = 1 -> ( -. ( # ` V ) = 1 -> ( K = 0 \/ K = 2 ) ) ) ) ) )
66	34, 65	syl 17	. . . . . . . . . . . . . . 15 |- ( G RegUSGraph K -> ( G e. FriendGraph -> ( ( V e. Fin /\ V =/= (/) ) -> ( K = 1 -> ( -. ( # ` V ) = 1 -> ( K = 0 \/ K = 2 ) ) ) ) ) )
67	66	impcom 446	. . . . . . . . . . . . . 14 |- ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( ( V e. Fin /\ V =/= (/) ) -> ( K = 1 -> ( -. ( # ` V ) = 1 -> ( K = 0 \/ K = 2 ) ) ) ) )
68	67	impcom 446	. . . . . . . . . . . . 13 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 1 -> ( -. ( # ` V ) = 1 -> ( K = 0 \/ K = 2 ) ) ) )
69	68	com13 88	. . . . . . . . . . . 12 |- ( -. ( # ` V ) = 1 -> ( K = 1 -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) ) )
70	32, 69	pm2.61i 176	. . . . . . . . . . 11 |- ( K = 1 -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) )
71	22, 70	syl6 35	. . . . . . . . . 10 |- ( ( ( K =/= 0 /\ K =/= 2 ) /\ K e. NN0 ) -> ( -. 1 < K -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) ) )
72	71	ex 450	. . . . . . . . 9 |- ( ( K =/= 0 /\ K =/= 2 ) -> ( K e. NN0 -> ( -. 1 < K -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) ) ) )
73	72	com23 86	. . . . . . . 8 |- ( ( K =/= 0 /\ K =/= 2 ) -> ( -. 1 < K -> ( K e. NN0 -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) ) ) )
74	9, 73	sylbir 225	. . . . . . 7 |- ( -. ( K = 0 \/ K = 2 ) -> ( -. 1 < K -> ( K e. NN0 -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) ) ) )
75	74	impcom 446	. . . . . 6 |- ( ( -. 1 < K /\ -. ( K = 0 \/ K = 2 ) ) -> ( K e. NN0 -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) ) )
76	75	com13 88	. . . . 5 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K e. NN0 -> ( ( -. 1 < K /\ -. ( K = 0 \/ K = 2 ) ) -> ( K = 0 \/ K = 2 ) ) ) )
77	8, 76	mpd 15	. . . 4 |- ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( ( -. 1 < K /\ -. ( K = 0 \/ K = 2 ) ) -> ( K = 0 \/ K = 2 ) ) )
78	77	com12 32	. . 3 |- ( ( -. 1 < K /\ -. ( K = 0 \/ K = 2 ) ) -> ( ( ( V e. Fin /\ V =/= (/) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) ) )
79	78	exp4b 632	. 2 |- ( -. 1 < K -> ( -. ( K = 0 \/ K = 2 ) -> ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( K = 0 \/ K = 2 ) ) ) ) )
80		simprl 794	. . . . 5 |- ( ( ( 1 < K /\ ( V e. Fin /\ V =/= (/) ) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> G e. FriendGraph )
81		simpl 473	. . . . . 6 |- ( ( V e. Fin /\ V =/= (/) ) -> V e. Fin )
82	81	ad2antlr 763	. . . . 5 |- ( ( ( 1 < K /\ ( V e. Fin /\ V =/= (/) ) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> V e. Fin )
83		simpr 477	. . . . . 6 |- ( ( V e. Fin /\ V =/= (/) ) -> V =/= (/) )
84	83	ad2antlr 763	. . . . 5 |- ( ( ( 1 < K /\ ( V e. Fin /\ V =/= (/) ) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> V =/= (/) )
85		simpl 473	. . . . . 6 |- ( ( 1 < K /\ ( V e. Fin /\ V =/= (/) ) ) -> 1 < K )
86	85, 26	anim12ci 591	. . . . 5 |- ( ( ( 1 < K /\ ( V e. Fin /\ V =/= (/) ) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( G RegUSGraph K /\ 1 < K ) )
87	6	frgrreggt1 27251	. . . . . 6 |- ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( ( G RegUSGraph K /\ 1 < K ) -> K = 2 ) )
88	87	imp 445	. . . . 5 |- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ ( G RegUSGraph K /\ 1 < K ) ) -> K = 2 )
89	80, 82, 84, 86, 88	syl31anc 1329	. . . 4 |- ( ( ( 1 < K /\ ( V e. Fin /\ V =/= (/) ) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> K = 2 )
90	89	olcd 408	. . 3 |- ( ( ( 1 < K /\ ( V e. Fin /\ V =/= (/) ) ) /\ ( G e. FriendGraph /\ G RegUSGraph K ) ) -> ( K = 0 \/ K = 2 ) )
91	90	exp31 630	. 2 |- ( 1 < K -> ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( K = 0 \/ K = 2 ) ) ) )
92		2a1 28	. 2 |- ( ( K = 0 \/ K = 2 ) -> ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( K = 0 \/ K = 2 ) ) ) )
93	79, 91, 92	pm2.61ii 177	1 |- ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( K = 0 \/ K = 2 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   class class class wbr 4653   ` cfv 5888   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292  NN0*cxnn0 11363   #chash 13117  Vtxcvtx 25874   USGraph cusgr 26044  VtxDegcvtxdg 26361   RegUSGraph crusgr 26452   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-reps 13306  df-csh 13535  df-s2 13593  df-s3 13594  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-vtxdg 26362  df-rgr 26453  df-rusgr 26454  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726  df-clwwlks 26877  df-clwwlksn 26878  df-frgr 27121

This theorem is referenced by:  frgrregord013  27253