Theorem 1egrvtxdg0 26407

Description: The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)

Hypotheses

Ref	Expression
1egrvtxdg1.v	|- ( ph -> (Vtx ` G ) = V )
1egrvtxdg1.a	|- ( ph -> A e. X )
1egrvtxdg1.b	|- ( ph -> B e. V )
1egrvtxdg1.c	|- ( ph -> C e. V )
1egrvtxdg1.n	|- ( ph -> B =/= C )
1egrvtxdg0.d	|- ( ph -> D e. V )
1egrvtxdg0.n	|- ( ph -> C =/= D )
1egrvtxdg0.i	|- ( ph -> (iEdg ` G ) = { <. A , { B , D } >. } )

Assertion

Ref	Expression
1egrvtxdg0	|- ( ph -> ( (VtxDeg ` G ) ` C ) = 0 )

Proof of Theorem 1egrvtxdg0

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1egrvtxdg1.v	. . . . 5 |- ( ph -> (Vtx ` G ) = V )
2	1	adantl 482	. . . 4 |- ( ( B = D /\ ph ) -> (Vtx ` G ) = V )
3		1egrvtxdg1.a	. . . . 5 |- ( ph -> A e. X )
4	3	adantl 482	. . . 4 |- ( ( B = D /\ ph ) -> A e. X )
5		1egrvtxdg1.b	. . . . 5 |- ( ph -> B e. V )
6	5	adantl 482	. . . 4 |- ( ( B = D /\ ph ) -> B e. V )
7		1egrvtxdg0.i	. . . . . 6 |- ( ph -> (iEdg ` G ) = { <. A , { B , D } >. } )
8	7	adantl 482	. . . . 5 |- ( ( B = D /\ ph ) -> (iEdg ` G ) = { <. A , { B , D } >. } )
9		preq2 4269	. . . . . . . . . 10 |- ( D = B -> { B , D } = { B , B } )
10	9	eqcoms 2630	. . . . . . . . 9 |- ( B = D -> { B , D } = { B , B } )
11		dfsn2 4190	. . . . . . . . 9 |- { B } = { B , B }
12	10, 11	syl6eqr 2674	. . . . . . . 8 |- ( B = D -> { B , D } = { B } )
13	12	adantr 481	. . . . . . 7 |- ( ( B = D /\ ph ) -> { B , D } = { B } )
14	13	opeq2d 4409	. . . . . 6 |- ( ( B = D /\ ph ) -> <. A , { B , D } >. = <. A , { B } >. )
15	14	sneqd 4189	. . . . 5 |- ( ( B = D /\ ph ) -> { <. A , { B , D } >. } = { <. A , { B } >. } )
16	8, 15	eqtrd 2656	. . . 4 |- ( ( B = D /\ ph ) -> (iEdg ` G ) = { <. A , { B } >. } )
17		1egrvtxdg1.c	. . . . . . 7 |- ( ph -> C e. V )
18		1egrvtxdg1.n	. . . . . . . 8 |- ( ph -> B =/= C )
19	18	necomd 2849	. . . . . . 7 |- ( ph -> C =/= B )
20	17, 19	jca 554	. . . . . 6 |- ( ph -> ( C e. V /\ C =/= B ) )
21		eldifsn 4317	. . . . . 6 |- ( C e. ( V \ { B } ) <-> ( C e. V /\ C =/= B ) )
22	20, 21	sylibr 224	. . . . 5 |- ( ph -> C e. ( V \ { B } ) )
23	22	adantl 482	. . . 4 |- ( ( B = D /\ ph ) -> C e. ( V \ { B } ) )
24	2, 4, 6, 16, 23	1loopgrvd0 26400	. . 3 |- ( ( B = D /\ ph ) -> ( (VtxDeg ` G ) ` C ) = 0 )
25	24	ex 450	. 2 |- ( B = D -> ( ph -> ( (VtxDeg ` G ) ` C ) = 0 ) )
26		necom 2847	. . . . . . . . . 10 |- ( B =/= C <-> C =/= B )
27		df-ne 2795	. . . . . . . . . 10 |- ( C =/= B <-> -. C = B )
28	26, 27	sylbb 209	. . . . . . . . 9 |- ( B =/= C -> -. C = B )
29	18, 28	syl 17	. . . . . . . 8 |- ( ph -> -. C = B )
30		1egrvtxdg0.n	. . . . . . . . 9 |- ( ph -> C =/= D )
31	30	neneqd 2799	. . . . . . . 8 |- ( ph -> -. C = D )
32	29, 31	jca 554	. . . . . . 7 |- ( ph -> ( -. C = B /\ -. C = D ) )
33	32	adantl 482	. . . . . 6 |- ( ( B =/= D /\ ph ) -> ( -. C = B /\ -. C = D ) )
34		ioran 511	. . . . . 6 |- ( -. ( C = B \/ C = D ) <-> ( -. C = B /\ -. C = D ) )
35	33, 34	sylibr 224	. . . . 5 |- ( ( B =/= D /\ ph ) -> -. ( C = B \/ C = D ) )
36		edgval 25941	. . . . . . . . 9 |- (Edg ` G ) = ran (iEdg ` G )
37	7	rneqd 5353	. . . . . . . . . 10 |- ( ph -> ran (iEdg ` G ) = ran { <. A , { B , D } >. } )
38		rnsnopg 5614	. . . . . . . . . . 11 |- ( A e. X -> ran { <. A , { B , D } >. } = { { B , D } } )
39	3, 38	syl 17	. . . . . . . . . 10 |- ( ph -> ran { <. A , { B , D } >. } = { { B , D } } )
40	37, 39	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ran (iEdg ` G ) = { { B , D } } )
41	36, 40	syl5eq 2668	. . . . . . . 8 |- ( ph -> (Edg ` G ) = { { B , D } } )
42	41	adantl 482	. . . . . . 7 |- ( ( B =/= D /\ ph ) -> (Edg ` G ) = { { B , D } } )
43	42	rexeqdv 3145	. . . . . 6 |- ( ( B =/= D /\ ph ) -> ( E. e e. (Edg ` G ) C e. e <-> E. e e. { { B , D } } C e. e ) )
44		prex 4909	. . . . . . 7 |- { B , D } e. _V
45		eleq2 2690	. . . . . . . 8 |- ( e = { B , D } -> ( C e. e <-> C e. { B , D } ) )
46	45	rexsng 4219	. . . . . . 7 |- ( { B , D } e. _V -> ( E. e e. { { B , D } } C e. e <-> C e. { B , D } ) )
47	44, 46	mp1i 13	. . . . . 6 |- ( ( B =/= D /\ ph ) -> ( E. e e. { { B , D } } C e. e <-> C e. { B , D } ) )
48		elprg 4196	. . . . . . . 8 |- ( C e. V -> ( C e. { B , D } <-> ( C = B \/ C = D ) ) )
49	17, 48	syl 17	. . . . . . 7 |- ( ph -> ( C e. { B , D } <-> ( C = B \/ C = D ) ) )
50	49	adantl 482	. . . . . 6 |- ( ( B =/= D /\ ph ) -> ( C e. { B , D } <-> ( C = B \/ C = D ) ) )
51	43, 47, 50	3bitrd 294	. . . . 5 |- ( ( B =/= D /\ ph ) -> ( E. e e. (Edg ` G ) C e. e <-> ( C = B \/ C = D ) ) )
52	35, 51	mtbird 315	. . . 4 |- ( ( B =/= D /\ ph ) -> -. E. e e. (Edg ` G ) C e. e )
53		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
54	3	adantl 482	. . . . . 6 |- ( ( B =/= D /\ ph ) -> A e. X )
55	5, 1	eleqtrrd 2704	. . . . . . 7 |- ( ph -> B e. (Vtx ` G ) )
56	55	adantl 482	. . . . . 6 |- ( ( B =/= D /\ ph ) -> B e. (Vtx ` G ) )
57		1egrvtxdg0.d	. . . . . . . 8 |- ( ph -> D e. V )
58	57, 1	eleqtrrd 2704	. . . . . . 7 |- ( ph -> D e. (Vtx ` G ) )
59	58	adantl 482	. . . . . 6 |- ( ( B =/= D /\ ph ) -> D e. (Vtx ` G ) )
60	7	adantl 482	. . . . . 6 |- ( ( B =/= D /\ ph ) -> (iEdg ` G ) = { <. A , { B , D } >. } )
61		simpl 473	. . . . . 6 |- ( ( B =/= D /\ ph ) -> B =/= D )
62	53, 54, 56, 59, 60, 61	usgr1e 26137	. . . . 5 |- ( ( B =/= D /\ ph ) -> G e. USGraph )
63	17, 1	eleqtrrd 2704	. . . . . 6 |- ( ph -> C e. (Vtx ` G ) )
64	63	adantl 482	. . . . 5 |- ( ( B =/= D /\ ph ) -> C e. (Vtx ` G ) )
65		eqid 2622	. . . . . 6 |- (Edg ` G ) = (Edg ` G )
66		eqid 2622	. . . . . 6 |- (VtxDeg ` G ) = (VtxDeg ` G )
67	53, 65, 66	vtxdusgr0edgnel 26391	. . . . 5 |- ( ( G e. USGraph /\ C e. (Vtx ` G ) ) -> ( ( (VtxDeg ` G ) ` C ) = 0 <-> -. E. e e. (Edg ` G ) C e. e ) )
68	62, 64, 67	syl2anc 693	. . . 4 |- ( ( B =/= D /\ ph ) -> ( ( (VtxDeg ` G ) ` C ) = 0 <-> -. E. e e. (Edg ` G ) C e. e ) )
69	52, 68	mpbird 247	. . 3 |- ( ( B =/= D /\ ph ) -> ( (VtxDeg ` G ) ` C ) = 0 )
70	69	ex 450	. 2 |- ( B =/= D -> ( ph -> ( (VtxDeg ` G ) ` C ) = 0 ) )
71	25, 70	pm2.61ine 2877	1 |- ( ph -> ( (VtxDeg ` G ) ` C ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   {csn 4177   {cpr 4179   <.cop 4183   ran crn 5115   ` cfv 5888   0cc0 9936  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   USGraph cusgr 26044  VtxDegcvtxdg 26361

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-uspgr 26045  df-usgr 26046  df-vtxdg 26362

This theorem is referenced by: (None)