Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rusgr1vtx | Structured version Visualization version GIF version |
Description: If a k-regular simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
rusgr1vtx | ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbgr1vtx 27418 | . . . 4 ⊢ ((♯‘(Vtx‘𝐺)) = 1 → (𝐺 NeighbVtx 𝑣) = ∅) | |
2 | 1 | ralrimivw 3099 | . . 3 ⊢ ((♯‘(Vtx‘𝐺)) = 1 → ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅) |
3 | eqid 2734 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
4 | 3 | rusgrpropnb 27643 | . . 3 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)) |
5 | 2, 4 | anim12i 616 | . 2 ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅ ∧ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))) |
6 | fvex 6719 | . . . . . . . 8 ⊢ (Vtx‘𝐺) ∈ V | |
7 | rusgr1vtxlem 27647 | . . . . . . . . 9 ⊢ (((∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅) ∧ ((Vtx‘𝐺) ∈ V ∧ (♯‘(Vtx‘𝐺)) = 1)) → 𝐾 = 0) | |
8 | 7 | ex 416 | . . . . . . . 8 ⊢ ((∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅) → (((Vtx‘𝐺) ∈ V ∧ (♯‘(Vtx‘𝐺)) = 1) → 𝐾 = 0)) |
9 | 6, 8 | mpani 696 | . . . . . . 7 ⊢ ((∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅) → ((♯‘(Vtx‘𝐺)) = 1 → 𝐾 = 0)) |
10 | 9 | ex 416 | . . . . . 6 ⊢ (∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾 → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅ → ((♯‘(Vtx‘𝐺)) = 1 → 𝐾 = 0))) |
11 | 10 | 3ad2ant3 1137 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅ → ((♯‘(Vtx‘𝐺)) = 1 → 𝐾 = 0))) |
12 | 11 | com13 88 | . . . 4 ⊢ ((♯‘(Vtx‘𝐺)) = 1 → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅ → ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾) → 𝐾 = 0))) |
13 | 12 | impd 414 | . . 3 ⊢ ((♯‘(Vtx‘𝐺)) = 1 → ((∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅ ∧ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)) → 𝐾 = 0)) |
14 | 13 | adantr 484 | . 2 ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → ((∀𝑣 ∈ (Vtx‘𝐺)(𝐺 NeighbVtx 𝑣) = ∅ ∧ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝐺)(♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)) → 𝐾 = 0)) |
15 | 5, 14 | mpd 15 | 1 ⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3054 Vcvv 3401 ∅c0 4227 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 0cc0 10712 1c1 10713 ℕ0*cxnn0 12145 ♯chash 13879 Vtxcvtx 27059 USGraphcusgr 27212 NeighbVtx cnbgr 27392 RegUSGraph crusgr 27616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-xadd 12688 df-fz 13079 df-hash 13880 df-edg 27111 df-uhgr 27121 df-ushgr 27122 df-upgr 27145 df-umgr 27146 df-uspgr 27213 df-usgr 27214 df-nbgr 27393 df-vtxdg 27526 df-rgr 27617 df-rusgr 27618 |
This theorem is referenced by: frgrreg 28449 |
Copyright terms: Public domain | W3C validator |