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Theorem vdn0conngrumgrv2 30078
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 2725 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2725 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4 eqid 2725 . . . 4 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
51, 2, 3, 4vtxdumgrval 29372 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
65ad2ant2lr 746 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
7 umgruhgr 28989 . . . . . . . 8 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
82uhgrfun 28951 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9 funfn 6584 . . . . . . . . 9 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
109biimpi 215 . . . . . . . 8 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
117, 8, 103syl 18 . . . . . . 7 (𝐺 ∈ UMGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1211adantl 480 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1312adantr 479 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14 simpl 481 . . . . . . 7 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → 𝐺 ∈ ConnGraph)
1514adantr 479 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 𝐺 ∈ ConnGraph)
16 simpl 481 . . . . . . 7 ((𝑁𝑉 ∧ 1 < (♯‘𝑉)) → 𝑁𝑉)
1716adantl 480 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 𝑁𝑉)
18 simprr 771 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 1 < (♯‘𝑉))
191, 2conngrv2edg 30077 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝑁𝑉 ∧ 1 < (♯‘𝑉)) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
2015, 17, 18, 19syl3anc 1368 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒)
21 eleq2 2814 . . . . . . 7 (𝑒 = ((iEdg‘𝐺)‘𝑥) → (𝑁𝑒𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2221rexrn 7096 . . . . . 6 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒 ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2322biimpd 228 . . . . 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 3062 . . . 4 (∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2624, 25sylib 217 . . 3 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
27 fvex 6909 . . . . . . . 8 (iEdg‘𝐺) ∈ V
2827dmex 7917 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
2928a1i 11 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → dom (iEdg‘𝐺) ∈ V)
30 rabexg 5334 . . . . . 6 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
31 hasheq0 14358 . . . . . 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 4386 . . . . 5 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
3432, 33bitrdi 286 . . . 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 2966 . . 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 256 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0)
376, 36eqnetrd 2997 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 205  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  Vcvv 3461  c0 4322   class class class wbr 5149  dom cdm 5678  ran crn 5679  Fun wfun 6543   Fn wfn 6544  cfv 6549  0cc0 11140  1c1 11141   < clt 11280  chash 14325  Vtxcvtx 28881  iEdgciedg 28882  UHGraphcuhgr 28941  UMGraphcumgr 28966  VtxDegcvtxdg 29351  ConnGraphcconngr 30068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-n0 12506  df-xnn0 12578  df-z 12592  df-uz 12856  df-xadd 13128  df-fz 13520  df-fzo 13663  df-hash 14326  df-word 14501  df-uhgr 28943  df-upgr 28967  df-umgr 28968  df-vtxdg 29352  df-wlks 29485  df-wlkson 29486  df-trlson 29579  df-pthson 29604  df-conngr 30069
This theorem is referenced by:  vdgn0frgrv2  30177
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