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Mirrors > Home > MPE Home > Th. List > frgrregord13 | Structured version Visualization version GIF version |
Description: If a nonempty finite friendship graph is 𝐾-regular, then it must have order 1 or 3. Special case of frgrregord013 28332. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.) |
Ref | Expression |
---|---|
frgrreggt1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
frgrregord13 | ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1192 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FriendGraph ) | |
2 | simpl2 1193 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ∈ Fin) | |
3 | simpr 488 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾) | |
4 | frgrreggt1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | frgrregord013 28332 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) |
6 | 1, 2, 3, 5 | syl3anc 1372 | . 2 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) |
7 | hasheq0 13817 | . . . . . . . . 9 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅)) | |
8 | eqneqall 2945 | . . . . . . . . 9 ⊢ (𝑉 = ∅ → (𝑉 ≠ ∅ → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))) | |
9 | 7, 8 | syl6bi 256 | . . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 → (𝑉 ≠ ∅ → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))) |
10 | 9 | com23 86 | . . . . . . 7 ⊢ (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((♯‘𝑉) = 0 → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)))) |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝐺 ∈ FriendGraph → (𝑉 ∈ Fin → (𝑉 ≠ ∅ → ((♯‘𝑉) = 0 → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))))) |
12 | 11 | 3imp 1112 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((♯‘𝑉) = 0 → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))) |
13 | 12 | adantr 484 | . . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))) |
14 | 13 | com12 32 | . . 3 ⊢ ((♯‘𝑉) = 0 → (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))) |
15 | orc 866 | . . . 4 ⊢ ((♯‘𝑉) = 1 → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) | |
16 | 15 | a1d 25 | . . 3 ⊢ ((♯‘𝑉) = 1 → (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))) |
17 | olc 867 | . . . 4 ⊢ ((♯‘𝑉) = 3 → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) | |
18 | 17 | a1d 25 | . . 3 ⊢ ((♯‘𝑉) = 3 → (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))) |
19 | 14, 16, 18 | 3jaoi 1428 | . 2 ⊢ (((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3) → (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))) |
20 | 6, 19 | mpcom 38 | 1 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 ∨ w3o 1087 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∅c0 4212 class class class wbr 5031 ‘cfv 6340 Fincfn 8556 0cc0 10616 1c1 10617 3c3 11773 ♯chash 13783 Vtxcvtx 26941 RegUSGraph crusgr 27498 FriendGraph cfrgr 28195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-inf2 9178 ax-ac2 9964 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 ax-pre-sup 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-disj 4997 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-2o 8133 df-oadd 8136 df-er 8321 df-ec 8323 df-qs 8327 df-map 8440 df-pm 8441 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-sup 8980 df-inf 8981 df-oi 9048 df-dju 9404 df-card 9442 df-ac 9617 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-div 11377 df-nn 11718 df-2 11780 df-3 11781 df-n0 11978 df-xnn0 12050 df-z 12064 df-uz 12326 df-rp 12474 df-xadd 12592 df-ico 12828 df-fz 12983 df-fzo 13126 df-fl 13254 df-mod 13330 df-seq 13462 df-exp 13523 df-hash 13784 df-word 13957 df-lsw 14005 df-concat 14013 df-s1 14040 df-substr 14093 df-pfx 14123 df-reps 14221 df-csh 14241 df-s2 14300 df-s3 14301 df-cj 14549 df-re 14550 df-im 14551 df-sqrt 14685 df-abs 14686 df-clim 14936 df-sum 15137 df-dvds 15701 df-gcd 15939 df-prm 16114 df-phi 16204 df-vtx 26943 df-iedg 26944 df-edg 26993 df-uhgr 27003 df-ushgr 27004 df-upgr 27027 df-umgr 27028 df-uspgr 27095 df-usgr 27096 df-fusgr 27259 df-nbgr 27275 df-vtxdg 27408 df-rgr 27499 df-rusgr 27500 df-wlks 27541 df-wlkson 27542 df-trls 27634 df-trlson 27635 df-pths 27657 df-spths 27658 df-pthson 27659 df-spthson 27660 df-wwlks 27768 df-wwlksn 27769 df-wwlksnon 27770 df-wspthsn 27771 df-wspthsnon 27772 df-clwwlk 27919 df-clwwlkn 27962 df-clwwlknon 28025 df-conngr 28124 df-frgr 28196 |
This theorem is referenced by: frgrogt3nreg 28334 |
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