Theorem vdn0conngrumgrv2 30197

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 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3		eqid 2733	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4		eqid 2733	. . . 4 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
5	1, 2, 3, 4	vtxdumgrval 29486	. . 3 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
6	5	ad2ant2lr 748	. 2 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
7		umgruhgr 29103	. . . . . . . 8 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
8	2	uhgrfun 29065	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9		funfn 6519	. . . . . . . . 9 ⊢ (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
10	9	biimpi 216	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
13	12	adantr 480	. . . . 5 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14		simpl 482	. . . . . . 7 ⊢ ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → 𝐺 ∈ ConnGraph)
15	14	adantr 480	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → 𝐺 ∈ ConnGraph)
16		simpl 482	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉)) → 𝑁 ∈ 𝑉)
17	16	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → 𝑁 ∈ 𝑉)
18		simprr 772	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → 1 < (♯‘𝑉))
19	1, 2	conngrv2edg 30196	. . . . . 6 ⊢ ((𝐺 ∈ ConnGraph ∧ 𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉)) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁 ∈ 𝑒)
20	15, 17, 18, 19	syl3anc 1373	. . . . 5 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁 ∈ 𝑒)
21		eleq2 2822	. . . . . . 7 ⊢ (𝑒 = ((iEdg‘𝐺)‘𝑥) → (𝑁 ∈ 𝑒 ↔ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
22	21	rexrn 7029	. . . . . 6 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁 ∈ 𝑒 ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
23	22	biimpd 229	. . . . 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 3060	. . . 4 ⊢ (∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
26	24, 25	sylib 218	. . 3 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
27		fvex 6844	. . . . . . . 8 ⊢ (iEdg‘𝐺) ∈ V
28	27	dmex 7848	. . . . . . 7 ⊢ dom (iEdg‘𝐺) ∈ V
29	28	a1i 11	. . . . . 6 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → dom (iEdg‘𝐺) ∈ V)
30		rabexg 5279	. . . . . 6 ⊢ (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
31		hasheq0 14277	. . . . . 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 4337	. . . . 5 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
34	32, 33	bitrdi 287	. . . 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 2965	. . 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 257	. 2 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0)
37	6, 36	eqnetrd 2996	1 ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  {crab 3396  Vcvv 3437  ∅c0 4282   class class class wbr 5095  dom cdm 5621  ran crn 5622  Fun wfun 6483   Fn wfn 6484  ‘cfv 6489  0cc0 11017  1c1 11018   < clt 11157  ♯chash 14244  Vtxcvtx 28995  iEdgciedg 28996  UHGraphcuhgr 29055  UMGraphcumgr 29080  VtxDegcvtxdg 29465  ConnGraphcconngr 30187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-er 8631  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-n0 12393  df-xnn0 12466  df-z 12480  df-uz 12743  df-xadd 13018  df-fz 13415  df-fzo 13562  df-hash 14245  df-word 14428  df-uhgr 29057  df-upgr 29081  df-umgr 29082  df-vtxdg 29466  df-wlks 29599  df-wlkson 29600  df-trlson 29691  df-pthson 29715  df-conngr 30188

This theorem is referenced by:  vdgn0frgrv2  30296