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Theorem vdn0conngrumgrv2 27373
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 2770 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2770 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4 eqid 2770 . . . 4 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
51, 2, 3, 4vtxdumgrval 26616 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
65ad2ant2lr 734 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
7 umgruhgr 26219 . . . . . . . 8 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
82uhgrfun 26181 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9 funfn 6061 . . . . . . . . 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 467 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1312adantr 466 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14 simpl 468 . . . . . . 7 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → 𝐺 ∈ ConnGraph)
1514adantr 466 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 𝐺 ∈ ConnGraph)
16 simpl 468 . . . . . . 7 ((𝑁𝑉 ∧ 1 < (♯‘𝑉)) → 𝑁𝑉)
1716adantl 467 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 𝑁𝑉)
18 simprr 748 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 1 < (♯‘𝑉))
191, 2conngrv2edg 27372 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝑁𝑉 ∧ 1 < (♯‘𝑉)) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
2015, 17, 18, 19syl3anc 1475 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
21 eleq2 2838 . . . . . . 7 (𝑒 = ((iEdg‘𝐺)‘𝑥) → (𝑁𝑒𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2221rexrn 6504 . . . . . 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 3143 . . . 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 6342 . . . . . . . 8 (iEdg‘𝐺) ∈ V
2827dmex 7245 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
2928a1i 11 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → dom (iEdg‘𝐺) ∈ V)
30 rabexg 4942 . . . . . 6 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
31 hasheq0 13355 . . . . . 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 4101 . . . . 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 2978 . . 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 3009 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 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  c0 4061   class class class wbr 4784  dom cdm 5249  ran crn 5250  Fun wfun 6025   Fn wfn 6026  cfv 6031  0cc0 10137  1c1 10138   < clt 10275  chash 13320  Vtxcvtx 26094  iEdgciedg 26095  UHGraphcuhgr 26171  UMGraphcumgr 26196  VtxDegcvtxdg 26595  ConnGraphcconngr 27363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ifp 1049  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-xadd 12151  df-fz 12533  df-fzo 12673  df-hash 13321  df-word 13494  df-uhgr 26173  df-upgr 26197  df-umgr 26198  df-vtxdg 26596  df-wlks 26729  df-wlkson 26730  df-trls 26823  df-trlson 26824  df-pths 26846  df-pthson 26848  df-conngr 27364
This theorem is referenced by:  vdgn0frgrv2  27474
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