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Theorem vdn0conngrumgrv2 27960
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 2821 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
3 eqid 2821 . . . 4 dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
4 eqid 2821 . . . 4 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
51, 2, 3, 4vtxdumgrval 27255 . . 3 ((𝐺 ∈ UMGraph ∧ 𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
65ad2ant2lr 747 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) = (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}))
7 umgruhgr 26876 . . . . . . . 8 (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph)
82uhgrfun 26838 . . . . . . . 8 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9 funfn 6358 . . . . . . . . 9 (Fun (iEdg‘𝐺) ↔ (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
109biimpi 219 . . . . . . . 8 (Fun (iEdg‘𝐺) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
117, 8, 103syl 18 . . . . . . 7 (𝐺 ∈ UMGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1211adantl 485 . . . . . 6 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
1312adantr 484 . . . . 5 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
14 simpl 486 . . . . . . 7 ((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) → 𝐺 ∈ ConnGraph)
1514adantr 484 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 𝐺 ∈ ConnGraph)
16 simpl 486 . . . . . . 7 ((𝑁𝑉 ∧ 1 < (♯‘𝑉)) → 𝑁𝑉)
1716adantl 485 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 𝑁𝑉)
18 simprr 772 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → 1 < (♯‘𝑉))
191, 2conngrv2edg 27959 . . . . . 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 2900 . . . . . . 7 (𝑒 = ((iEdg‘𝐺)‘𝑥) → (𝑁𝑒𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2221rexrn 6826 . . . . . 6 ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → (∃𝑒 ∈ ran (iEdg‘𝐺)𝑁𝑒 ↔ ∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥)))
2322biimpd 232 . . . . 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 3227 . . . 4 (∃𝑥 ∈ dom (iEdg‘𝐺)𝑁 ∈ ((iEdg‘𝐺)‘𝑥) ↔ ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
2624, 25sylib 221 . . 3 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ¬ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
27 fvex 6656 . . . . . . . 8 (iEdg‘𝐺) ∈ V
2827dmex 7591 . . . . . . 7 dom (iEdg‘𝐺) ∈ V
2928a1i 11 . . . . . 6 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → dom (iEdg‘𝐺) ∈ V)
30 rabexg 5207 . . . . . 6 (dom (iEdg‘𝐺) ∈ V → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} ∈ V)
31 hasheq0 13708 . . . . . 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 4311 . . . . 5 ({𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥))
3432, 33syl6bb 290 . . . 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 3043 . . 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 260 . 2 (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑁 ∈ ((iEdg‘𝐺)‘𝑥)}) ≠ 0)
376, 36eqnetrd 3074 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 209  wa 399   = wceq 1538  wcel 2115  wne 3007  wral 3126  wrex 3127  {crab 3130  Vcvv 3471  c0 4266   class class class wbr 5039  dom cdm 5528  ran crn 5529  Fun wfun 6322   Fn wfn 6323  cfv 6328  0cc0 10514  1c1 10515   < clt 10652  chash 13674  Vtxcvtx 26768  iEdgciedg 26769  UHGraphcuhgr 26828  UMGraphcumgr 26853  VtxDegcvtxdg 27234  ConnGraphcconngr 27950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-n0 11876  df-xnn0 11946  df-z 11960  df-uz 12222  df-xadd 12486  df-fz 12876  df-fzo 13017  df-hash 13675  df-word 13846  df-uhgr 26830  df-upgr 26854  df-umgr 26855  df-vtxdg 27235  df-wlks 27368  df-wlkson 27369  df-trls 27461  df-trlson 27462  df-pths 27484  df-pthson 27486  df-conngr 27951
This theorem is referenced by:  vdgn0frgrv2  28059
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