Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  vdn0conngrumgrv2 Structured version   Visualization version   GIF version

Theorem vdn0conngrumgrv2 27036
 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 ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)

Proof of Theorem vdn0conngrumgrv2
Dummy variables 𝑒 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vdn0conngrv2.v . . . 4 𝑉 = (Vtx‘𝐺)
2 eqid 2620 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2620 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4 eqid 2620 . . . 4 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
51, 2, 3, 4vtxdumgrval 26363 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
65ad2ant2lr 783 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
7 umgruhgr 25980 . . . . . . . 8 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
82uhgrfun 25942 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9 funfn 5906 . . . . . . . . 9 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
109biimpi 206 . . . . . . . 8 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
117, 8, 103syl 18 . . . . . . 7 (𝐺 ∈ UMGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1211adantl 482 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1312adantr 481 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14 simpl 473 . . . . . . 7 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) → 𝐺 ∈ ConnGraph)
1514adantr 481 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → 𝐺 ∈ ConnGraph)
16 simpl 473 . . . . . . 7 ((𝑁𝑉 ∧ 1 < (#‘𝑉)) → 𝑁𝑉)
1716adantl 482 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → 𝑁𝑉)
18 simprr 795 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → 1 < (#‘𝑉))
191, 2conngrv2edg 27035 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝑁𝑉 ∧ 1 < (#‘𝑉)) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
2015, 17, 18, 19syl3anc 1324 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
21 eleq2 2688 . . . . . . 7 (𝑒 = ((iEdg‘𝐺)‘𝑥) → (𝑁𝑒𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2221rexrn 6347 . . . . . 6 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒 ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2322biimpd 219 . . . . 5 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒 → ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2413, 20, 23sylc 65 . . . 4 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
25 dfrex2 2993 . . . 4 (∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2624, 25sylib 208 . . 3 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
27 fvex 6188 . . . . . . . 8 (iEdg‘𝐺) ∈ V
2827dmex 7084 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
2928a1i 11 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → dom (iEdg‘𝐺) ∈ V)
30 rabexg 4803 . . . . . 6 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
31 hasheq0 13137 . . . . . 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)} = ∅))
3229, 30, 313syl 18 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0 ↔ {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅))
33 rabeq0 3948 . . . . 5 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
3432, 33syl6bb 276 . . . 4 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0 ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
3534necon3abid 2827 . . 3 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0 ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
3626, 35mpbird 247 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0)
376, 36eqnetrd 2858 1 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909  ∃wrex 2910  {crab 2913  Vcvv 3195  ∅c0 3907   class class class wbr 4644  dom cdm 5104  ran crn 5105  Fun wfun 5870   Fn wfn 5871  ‘cfv 5876  0cc0 9921  1c1 9922   < clt 10059  #chash 13100  Vtxcvtx 25855  iEdgciedg 25856   UHGraph cuhgr 25932   UMGraph cumgr 25957  VtxDegcvtxdg 26342  ConnGraphcconngr 27026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-xadd 11932  df-fz 12312  df-fzo 12450  df-hash 13101  df-word 13282  df-uhgr 25934  df-upgr 25958  df-umgr 25959  df-vtxdg 26343  df-wlks 26476  df-wlkson 26477  df-trls 26570  df-trlson 26571  df-pths 26593  df-pthson 26595  df-conngr 27027 This theorem is referenced by:  vdgn0frgrv2  27139
 Copyright terms: Public domain W3C validator