Theorem vdgn1frgrv2 27160

Description: Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 4-Apr-2021.)

Hypothesis

Ref	Expression
vdn1frgrv2.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
vdgn1frgrv2	|- ( ( G e. FriendGraph /\ N e. V ) -> ( 1 < ( # ` V ) -> ( (VtxDeg ` G ) ` N ) =/= 1 ) )

Proof of Theorem vdgn1frgrv2

Dummy variables a b c x i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrusgr 27124	. . . . . 6 |- ( G e. FriendGraph -> G e. USGraph )
2	1	anim1i 592	. . . . 5 |- ( ( G e. FriendGraph /\ N e. V ) -> ( G e. USGraph /\ N e. V ) )
3	2	adantr 481	. . . 4 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( G e. USGraph /\ N e. V ) )
4		vdn1frgrv2.v	. . . . 5 |- V = (Vtx ` G )
5		eqid 2622	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
6		eqid 2622	. . . . 5 |- dom (iEdg ` G ) = dom (iEdg ` G )
7		eqid 2622	. . . . 5 |- (VtxDeg ` G ) = (VtxDeg ` G )
8	4, 5, 6, 7	vtxdusgrval 26383	. . . 4 |- ( ( G e. USGraph /\ N e. V ) -> ( (VtxDeg ` G ) ` N ) = ( # ` { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } ) )
9	3, 8	syl 17	. . 3 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( (VtxDeg ` G ) ` N ) = ( # ` { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } ) )
10		eqid 2622	. . . . . . 7 |- (Edg ` G ) = (Edg ` G )
11	4, 10	3cyclfrgrrn2 27151	. . . . . 6 |- ( ( G e. FriendGraph /\ 1 < ( # ` V ) ) -> A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) )
12	11	adantlr 751	. . . . 5 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) )
13		preq1 4268	. . . . . . . . . . . . . . . 16 |- ( a = N -> { a , b } = { N , b } )
14	13	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( a = N -> ( { a , b } e. (Edg ` G ) <-> { N , b } e. (Edg ` G ) ) )
15		preq2 4269	. . . . . . . . . . . . . . . 16 |- ( a = N -> { c , a } = { c , N } )
16	15	eleq1d 2686	. . . . . . . . . . . . . . 15 |- ( a = N -> ( { c , a } e. (Edg ` G ) <-> { c , N } e. (Edg ` G ) ) )
17	14, 16	3anbi13d 1401	. . . . . . . . . . . . . 14 |- ( a = N -> ( ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) <-> ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) )
18	17	anbi2d 740	. . . . . . . . . . . . 13 |- ( a = N -> ( ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) <-> ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) ) )
19	18	2rexbidv 3057	. . . . . . . . . . . 12 |- ( a = N -> ( E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) <-> E. b e. V E. c e. V ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) ) )
20	19	rspcva 3307	. . . . . . . . . . 11 |- ( ( N e. V /\ A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) ) -> E. b e. V E. c e. V ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) )
21	1	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) /\ N e. V ) /\ G e. FriendGraph ) -> G e. USGraph )
22		simplll 798	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) /\ N e. V ) /\ G e. FriendGraph ) -> b =/= c )
23		3simpb 1059	. . . . . . . . . . . . . . . . . 18 |- ( ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) -> ( { N , b } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) )
24	23	ad3antlr 767	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) /\ N e. V ) /\ G e. FriendGraph ) -> ( { N , b } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) )
25	5, 10	usgr2edg1 26104	. . . . . . . . . . . . . . . . 17 |- ( ( ( G e. USGraph /\ b =/= c ) /\ ( { N , b } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) )
26	21, 22, 24, 25	syl21anc 1325	. . . . . . . . . . . . . . . 16 |- ( ( ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) /\ N e. V ) /\ G e. FriendGraph ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) )
27	26	a1d 25	. . . . . . . . . . . . . . 15 |- ( ( ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) /\ N e. V ) /\ G e. FriendGraph ) -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) )
28	27	ex 450	. . . . . . . . . . . . . 14 |- ( ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) /\ N e. V ) -> ( G e. FriendGraph -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) )
29	28	ex 450	. . . . . . . . . . . . 13 |- ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) -> ( N e. V -> ( G e. FriendGraph -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) )
30	29	a1i 11	. . . . . . . . . . . 12 |- ( ( b e. V /\ c e. V ) -> ( ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) -> ( N e. V -> ( G e. FriendGraph -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) ) )
31	30	rexlimivv 3036	. . . . . . . . . . 11 |- ( E. b e. V E. c e. V ( b =/= c /\ ( { N , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , N } e. (Edg ` G ) ) ) -> ( N e. V -> ( G e. FriendGraph -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) )
32	20, 31	syl 17	. . . . . . . . . 10 |- ( ( N e. V /\ A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) ) -> ( N e. V -> ( G e. FriendGraph -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) )
33	32	ex 450	. . . . . . . . 9 |- ( N e. V -> ( A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) -> ( N e. V -> ( G e. FriendGraph -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) ) )
34	33	pm2.43a 54	. . . . . . . 8 |- ( N e. V -> ( A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) -> ( G e. FriendGraph -> ( 1 < ( # ` V ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) )
35	34	com24 95	. . . . . . 7 |- ( N e. V -> ( 1 < ( # ` V ) -> ( G e. FriendGraph -> ( A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) )
36	35	com3r 87	. . . . . 6 |- ( G e. FriendGraph -> ( N e. V -> ( 1 < ( # ` V ) -> ( A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) ) ) )
37	36	imp31 448	. . . . 5 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( A. a e. V E. b e. V E. c e. V ( b =/= c /\ ( { a , b } e. (Edg ` G ) /\ { b , c } e. (Edg ` G ) /\ { c , a } e. (Edg ` G ) ) ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) )
38	12, 37	mpd 15	. . . 4 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) )
39		fvex 6201	. . . . . . . . 9 |- (iEdg ` G ) e. _V
40	39	dmex 7099	. . . . . . . 8 |- dom (iEdg ` G ) e. _V
41	40	a1i 11	. . . . . . 7 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> dom (iEdg ` G ) e. _V )
42		rabexg 4812	. . . . . . 7 |- ( dom (iEdg ` G ) e. _V -> { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } e. _V )
43		hash1snb 13207	. . . . . . 7 |- ( { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } e. _V -> ( ( # ` { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } ) = 1 <-> E. i { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } = { i } ) )
44	41, 42, 43	3syl 18	. . . . . 6 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( ( # ` { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } ) = 1 <-> E. i { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } = { i } ) )
45		reusn 4262	. . . . . 6 |- ( E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) <-> E. i { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } = { i } )
46	44, 45	syl6bbr 278	. . . . 5 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( ( # ` { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } ) = 1 <-> E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) )
47	46	necon3abid 2830	. . . 4 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( ( # ` { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } ) =/= 1 <-> -. E! x e. dom (iEdg ` G ) N e. ( (iEdg ` G ) ` x ) ) )
48	38, 47	mpbird 247	. . 3 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( # ` { x e. dom (iEdg ` G ) | N e. ( (iEdg ` G ) ` x ) } ) =/= 1 )
49	9, 48	eqnetrd 2861	. 2 |- ( ( ( G e. FriendGraph /\ N e. V ) /\ 1 < ( # ` V ) ) -> ( (VtxDeg ` G ) ` N ) =/= 1 )
50	49	ex 450	1 |- ( ( G e. FriendGraph /\ N e. V ) -> ( 1 < ( # ` V ) -> ( (VtxDeg ` G ) ` N ) =/= 1 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   {crab 2916   _Vcvv 3200   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   ` cfv 5888   1c1 9937   < clt 10074   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  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-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-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-upgr 25977  df-umgr 25978  df-usgr 26046  df-vtxdg 26362  df-frgr 27121

This theorem is referenced by:  vdgn1frgrv3  27161  vdgfrgrgt2  27162