Theorem vdn0conngrumgrv2 27056

Description: A vertex in a connected multigraph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)

Hypothesis

Ref	Expression
vdn0conngrv2.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
vdn0conngrumgrv2	⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)

Proof of Theorem vdn0conngrumgrv2

Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdn0conngrv2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2622	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2622	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4		eqid 2622	. . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
5	1, 2, 3, 4	vtxdumgrval 26382	. . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
6	5	ad2ant2lr 784	. 2 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
7		umgruhgr 25999	. . . . . . . 8 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
8	2	uhgrfun 25961	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9		funfn 5918	. . . . . . . . 9 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
10	9	biimpi 206	. . . . . . . 8 ⊢ (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
11	7, 8, 10	3syl 18	. . . . . . 7 ⊢ (𝐺 ∈ UMGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
12	11	adantl 482	. . . . . 6 ⊢ ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13	12	adantr 481	. . . . 5 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14		simpl 473	. . . . . . 7 ⊢ ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) → 𝐺 ∈ ConnGraph)
15	14	adantr 481	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → 𝐺 ∈ ConnGraph)
16		simpl 473	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉)) → 𝑁 ∈ 𝑉)
17	16	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → 𝑁 ∈ 𝑉)
18		simprr 796	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → 1 < (#‘𝑉))
19	1, 2	conngrv2edg 27055	. . . . . 6 ⊢ ((𝐺 ∈ ConnGraph ∧ 𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉)) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁 ∈ 𝑒)
20	15, 17, 18, 19	syl3anc 1326	. . . . 5 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁 ∈ 𝑒)
21		eleq2 2690	. . . . . . 7 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑥) → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
22	21	rexrn 6361	. . . . . 6 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁 ∈ 𝑒 ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
23	22	biimpd 219	. . . . 5 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁 ∈ 𝑒 → ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
24	13, 20, 23	sylc 65	. . . 4 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
25		dfrex2 2996	. . . 4 ⊢ (∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
26	24, 25	sylib 208	. . 3 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
27		fvex 6201	. . . . . . . 8 ⊢ (iEdg‘𝐺) ∈ V
28	27	dmex 7099	. . . . . . 7 ⊢ dom (iEdg‘𝐺) ∈ V
29	28	a1i 11	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → dom (iEdg‘𝐺) ∈ V)
30		rabexg 4812	. . . . . 6 ⊢ (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
31		hasheq0 13154	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0 ↔ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅))
32	29, 30, 31	3syl 18	. . . . 5 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0 ↔ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅))
33		rabeq0 3957	. . . . 5 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
34	32, 33	syl6bb 276	. . . 4 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0 ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
35	34	necon3abid 2830	. . 3 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0 ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
36	26, 35	mpbird 247	. 2 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0)
37	6, 36	eqnetrd 2861	1 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200  ∅c0 3915   class class class wbr 4653  dom cdm 5114  ran crn 5115  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  0cc0 9936  1c1 9937   < clt 10074  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951   UMGraph cumgr 25976  VtxDegcvtxdg 26361  ConnGraphcconngr 27046

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-ifp 1013  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-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-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-fzo 12466  df-hash 13118  df-word 13299  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-vtxdg 26362  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-pthson 26614  df-conngr 27047

This theorem is referenced by:  vdgn0frgrv2  27159