Theorem frgrwopreg 27187

Description: In a friendship graph there are either no vertices ( A = (/)) or exactly one vertex ( ( # ` A ) = 1) having degree K, or all ( B = (/)) or all except one vertices ( ( # ` B ) = 1) have degree K. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 3-Jan-2022.)

Hypotheses

Ref	Expression
frgrwopreg.v	|- V = (Vtx ` G )
frgrwopreg.d	|- D = (VtxDeg ` G )
frgrwopreg.a	|- A = { x e. V | ( D ` x ) = K }
frgrwopreg.b	|- B = ( V \ A )

Assertion

Ref	Expression
frgrwopreg	|- ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) )

Distinct variable groups:   x, V   x, A   x, G   x, K   x, D   x, B

Proof of Theorem frgrwopreg

Dummy variables a b y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrwopreg.v	. . 3 |- V = (Vtx ` G )
2		frgrwopreg.d	. . 3 |- D = (VtxDeg ` G )
3		frgrwopreg.a	. . 3 |- A = { x e. V | ( D ` x ) = K }
4		frgrwopreg.b	. . 3 |- B = ( V \ A )
5	1, 2, 3, 4	frgrwopreglem1 27176	. 2 |- ( A e. _V /\ B e. _V )
6		hashv01gt1 13133	. . . 4 |- ( A e. _V -> ( ( # ` A ) = 0 \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) )
7		hasheq0 13154	. . . . . 6 |- ( A e. _V -> ( ( # ` A ) = 0 <-> A = (/) ) )
8		biidd 252	. . . . . 6 |- ( A e. _V -> ( ( # ` A ) = 1 <-> ( # ` A ) = 1 ) )
9		biidd 252	. . . . . 6 |- ( A e. _V -> ( 1 < ( # ` A ) <-> 1 < ( # ` A ) ) )
10	7, 8, 9	3orbi123d 1398	. . . . 5 |- ( A e. _V -> ( ( ( # ` A ) = 0 \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) <-> ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) ) )
11		hashv01gt1 13133	. . . . . . 7 |- ( B e. _V -> ( ( # ` B ) = 0 \/ ( # ` B ) = 1 \/ 1 < ( # ` B ) ) )
12		hasheq0 13154	. . . . . . . . 9 |- ( B e. _V -> ( ( # ` B ) = 0 <-> B = (/) ) )
13		biidd 252	. . . . . . . . 9 |- ( B e. _V -> ( ( # ` B ) = 1 <-> ( # ` B ) = 1 ) )
14		biidd 252	. . . . . . . . 9 |- ( B e. _V -> ( 1 < ( # ` B ) <-> 1 < ( # ` B ) ) )
15	12, 13, 14	3orbi123d 1398	. . . . . . . 8 |- ( B e. _V -> ( ( ( # ` B ) = 0 \/ ( # ` B ) = 1 \/ 1 < ( # ` B ) ) <-> ( B = (/) \/ ( # ` B ) = 1 \/ 1 < ( # ` B ) ) ) )
16		olc 399	. . . . . . . . . . 11 |- ( B = (/) -> ( ( # ` B ) = 1 \/ B = (/) ) )
17	16	olcd 408	. . . . . . . . . 10 |- ( B = (/) -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) )
18	17	2a1d 26	. . . . . . . . 9 |- ( B = (/) -> ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
19		orc 400	. . . . . . . . . . 11 |- ( ( # ` B ) = 1 -> ( ( # ` B ) = 1 \/ B = (/) ) )
20	19	olcd 408	. . . . . . . . . 10 |- ( ( # ` B ) = 1 -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) )
21	20	2a1d 26	. . . . . . . . 9 |- ( ( # ` B ) = 1 -> ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
22		olc 399	. . . . . . . . . . . . 13 |- ( A = (/) -> ( ( # ` A ) = 1 \/ A = (/) ) )
23	22	orcd 407	. . . . . . . . . . . 12 |- ( A = (/) -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) )
24	23	2a1d 26	. . . . . . . . . . 11 |- ( A = (/) -> ( 1 < ( # ` B ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
25		orc 400	. . . . . . . . . . . . 13 |- ( ( # ` A ) = 1 -> ( ( # ` A ) = 1 \/ A = (/) ) )
26	25	orcd 407	. . . . . . . . . . . 12 |- ( ( # ` A ) = 1 -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) )
27	26	2a1d 26	. . . . . . . . . . 11 |- ( ( # ` A ) = 1 -> ( 1 < ( # ` B ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
28		eqid 2622	. . . . . . . . . . . . . . 15 |- (Edg ` G ) = (Edg ` G )
29	1, 2, 3, 4, 28	frgrwopreglem5 27185	. . . . . . . . . . . . . 14 |- ( ( G e. FriendGraph /\ 1 < ( # ` A ) /\ 1 < ( # ` B ) ) -> E. a e. A E. x e. A E. b e. B E. y e. B ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) )
30		frgrusgr 27124	. . . . . . . . . . . . . . . 16 |- ( G e. FriendGraph -> G e. USGraph )
31		simplll 798	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> G e. USGraph )
32		elrabi 3359	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( a e. { x e. V | ( D ` x ) = K } -> a e. V )
33	32, 3	eleq2s 2719	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( a e. A -> a e. V )
34	33	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( a e. A /\ x e. A ) -> a e. V )
35	34	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> a e. V )
36		rabidim1 3117	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x e. { x e. V | ( D ` x ) = K } -> x e. V )
37	36, 3	eleq2s 2719	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x e. A -> x e. V )
38	37	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( a e. A /\ x e. A ) -> x e. V )
39	38	ad3antlr 767	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> x e. V )
40		simprl 794	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> a =/= x )
41		eldifi 3732	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( b e. ( V \ A ) -> b e. V )
42	41, 4	eleq2s 2719	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( b e. B -> b e. V )
43	42	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( b e. B /\ y e. B ) -> b e. V )
44	43	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> b e. V )
45		eldifi 3732	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( y e. ( V \ A ) -> y e. V )
46	45, 4	eleq2s 2719	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( y e. B -> y e. V )
47	46	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( b e. B /\ y e. B ) -> y e. V )
48	47	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> y e. V )
49		simprr 796	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> b =/= y )
50	1, 28	4cyclusnfrgr 27156	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( G e. USGraph /\ ( a e. V /\ x e. V /\ a =/= x ) /\ ( b e. V /\ y e. V /\ b =/= y ) ) -> ( ( ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> G e/ FriendGraph ) )
51	31, 35, 39, 40, 44, 48, 49, 50	syl133anc 1349	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) /\ ( a =/= x /\ b =/= y ) ) -> ( ( ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> G e/ FriendGraph ) )
52	51	exp4b 632	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) -> ( ( a =/= x /\ b =/= y ) -> ( ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) -> ( ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) -> G e/ FriendGraph ) ) ) )
53	52	3impd 1281	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) -> ( ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> G e/ FriendGraph ) )
54		df-nel 2898	. . . . . . . . . . . . . . . . . . . . 21 |- ( G e/ FriendGraph <-> -. G e. FriendGraph )
55		pm2.21 120	. . . . . . . . . . . . . . . . . . . . 21 |- ( -. G e. FriendGraph -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) )
56	54, 55	sylbi 207	. . . . . . . . . . . . . . . . . . . 20 |- ( G e/ FriendGraph -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) )
57	53, 56	syl6 35	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) /\ ( b e. B /\ y e. B ) ) -> ( ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
58	57	rexlimdvva 3038	. . . . . . . . . . . . . . . . . 18 |- ( ( G e. USGraph /\ ( a e. A /\ x e. A ) ) -> ( E. b e. B E. y e. B ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
59	58	rexlimdvva 3038	. . . . . . . . . . . . . . . . 17 |- ( G e. USGraph -> ( E. a e. A E. x e. A E. b e. B E. y e. B ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
60	59	com23 86	. . . . . . . . . . . . . . . 16 |- ( G e. USGraph -> ( G e. FriendGraph -> ( E. a e. A E. x e. A E. b e. B E. y e. B ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
61	30, 60	mpcom 38	. . . . . . . . . . . . . . 15 |- ( G e. FriendGraph -> ( E. a e. A E. x e. A E. b e. B E. y e. B ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) )
62	61	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( G e. FriendGraph /\ 1 < ( # ` A ) /\ 1 < ( # ` B ) ) -> ( E. a e. A E. x e. A E. b e. B E. y e. B ( ( a =/= x /\ b =/= y ) /\ ( { a , b } e. (Edg ` G ) /\ { b , x } e. (Edg ` G ) ) /\ ( { x , y } e. (Edg ` G ) /\ { y , a } e. (Edg ` G ) ) ) -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) )
63	29, 62	mpd 15	. . . . . . . . . . . . 13 |- ( ( G e. FriendGraph /\ 1 < ( # ` A ) /\ 1 < ( # ` B ) ) -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) )
64	63	3exp 1264	. . . . . . . . . . . 12 |- ( G e. FriendGraph -> ( 1 < ( # ` A ) -> ( 1 < ( # ` B ) -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
65	64	com3l 89	. . . . . . . . . . 11 |- ( 1 < ( # ` A ) -> ( 1 < ( # ` B ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
66	24, 27, 65	3jaoi 1391	. . . . . . . . . 10 |- ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( 1 < ( # ` B ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
67	66	com12 32	. . . . . . . . 9 |- ( 1 < ( # ` B ) -> ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
68	18, 21, 67	3jaoi 1391	. . . . . . . 8 |- ( ( B = (/) \/ ( # ` B ) = 1 \/ 1 < ( # ` B ) ) -> ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
69	15, 68	syl6bi 243	. . . . . . 7 |- ( B e. _V -> ( ( ( # ` B ) = 0 \/ ( # ` B ) = 1 \/ 1 < ( # ` B ) ) -> ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) ) )
70	11, 69	mpd 15	. . . . . 6 |- ( B e. _V -> ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
71	70	com12 32	. . . . 5 |- ( ( A = (/) \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( B e. _V -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
72	10, 71	syl6bi 243	. . . 4 |- ( A e. _V -> ( ( ( # ` A ) = 0 \/ ( # ` A ) = 1 \/ 1 < ( # ` A ) ) -> ( B e. _V -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) ) )
73	6, 72	mpd 15	. . 3 |- ( A e. _V -> ( B e. _V -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) ) )
74	73	imp 445	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) ) )
75	5, 74	ax-mp 5	1 |- ( G e. FriendGraph -> ( ( ( # ` A ) = 1 \/ A = (/) ) \/ ( ( # ` B ) = 1 \/ B = (/) ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   {cpr 4179   class class class wbr 4653   ` cfv 5888   0cc0 9936   1c1 9937   < clt 10074   #chash 13117  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044  VtxDegcvtxdg 26361   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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-nbgr 26228  df-vtxdg 26362  df-frgr 27121

This theorem is referenced by:  frgrregorufr0  27188