Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vdn0conngrumgrv2 Structured version   Visualization version   GIF version

Theorem vdn0conngrumgrv2 41363
 Description: A vertex in a connected multigraph with more than one vertex cannot have degree 0. Formerly vdgn0frgrav2 26551. (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 2610 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2610 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4 eqid 2610 . . . 4 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
51, 2, 3, 4vtxdumgrval 40701 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
65ad2ant2lr 780 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) = (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
7 umgruhgr 25770 . . . . . . . 8 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
82uhgrfun 25732 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9 funfn 5833 . . . . . . . . 9 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
109biimpi 205 . . . . . . . 8 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
117, 8, 103syl 18 . . . . . . 7 (𝐺 ∈ UMGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1211adantl 481 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1312adantr 480 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14 simpl 472 . . . . . . 7 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) → 𝐺 ∈ ConnGraph)
1514adantr 480 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → 𝐺 ∈ ConnGraph)
16 simpl 472 . . . . . . 7 ((𝑁𝑉 ∧ 1 < (#‘𝑉)) → 𝑁𝑉)
1716adantl 481 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → 𝑁𝑉)
18 simprr 792 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → 1 < (#‘𝑉))
191, 2conngrv2edg 41362 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝑁𝑉 ∧ 1 < (#‘𝑉)) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
2015, 17, 18, 19syl3anc 1318 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
21 eleq2 2677 . . . . . . 7 (𝑒 = ((iEdg‘𝐺)‘𝑥) → (𝑁𝑒𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2221rexrn 6269 . . . . . 6 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒 ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2322biimpd 218 . . . . 5 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒 → ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2413, 20, 23sylc 63 . . . 4 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
25 dfrex2 2979 . . . 4 (∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2624, 25sylib 207 . . 3 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
27 fvex 6113 . . . . . . . 8 (iEdg‘𝐺) ∈ V
2827dmex 6991 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
2928a1i 11 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → dom (iEdg‘𝐺) ∈ V)
30 rabexg 4739 . . . . . 6 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
31 hasheq0 13015 . . . . . 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 3911 . . . . 5 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
3432, 33syl6bb 275 . . . 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 2818 . . 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 246 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0)
376, 36eqnetrd 2849 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ∅c0 3874   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  0cc0 9815  1c1 9816   < clt 9953  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   UMGraph cumgr 25748  VtxDegcvtxdg 40681  ConnGraphcconngr 41353 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-vtxdg 40682  df-1wlks 40800  df-wlkson 40802  df-trls 40901  df-trlson 40902  df-pths 40923  df-pthson 40925  df-conngr 41354 This theorem is referenced by:  vdgn0frgrv2  41465
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