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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 27253. (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 1064 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FriendGraph ) | |
2 | simpl2 1065 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ∈ Fin) | |
3 | simpr 477 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾) | |
4 | frgrreggt1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | frgrregord013 27253 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)) |
6 | 1, 2, 3, 5 | syl3anc 1326 | . 2 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)) |
7 | hasheq0 13154 | . . . . . . . . 9 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) = 0 ↔ 𝑉 = ∅)) | |
8 | eqneqall 2805 | . . . . . . . . 9 ⊢ (𝑉 = ∅ → (𝑉 ≠ ∅ → ((#‘𝑉) = 1 ∨ (#‘𝑉) = 3))) | |
9 | 7, 8 | syl6bi 243 | . . . . . . . 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 1256 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((#‘𝑉) = 0 → ((#‘𝑉) = 1 ∨ (#‘𝑉) = 3))) |
13 | 12 | adantr 481 | . . . 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 400 | . . . 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 399 | . . . 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 1391 | . 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 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 Fincfn 7955 0cc0 9936 1c1 9937 3c3 11071 #chash 13117 Vtxcvtx 25874 RegUSGraph crusgr 26452 FriendGraph cfrgr 27120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-reps 13306 df-csh 13535 df-s2 13593 df-s3 13594 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-phi 15471 df-vtx 25876 df-iedg 25877 df-edg 25940 df-uhgr 25953 df-ushgr 25954 df-upgr 25977 df-umgr 25978 df-uspgr 26045 df-usgr 26046 df-fusgr 26209 df-nbgr 26228 df-vtxdg 26362 df-rgr 26453 df-rusgr 26454 df-wlks 26495 df-wlkson 26496 df-trls 26589 df-trlson 26590 df-pths 26612 df-spths 26613 df-pthson 26614 df-spthson 26615 df-wwlks 26722 df-wwlksn 26723 df-wwlksnon 26724 df-wspthsn 26725 df-wspthsnon 26726 df-clwwlks 26877 df-clwwlksn 26878 df-conngr 27047 df-frgr 27121 |
This theorem is referenced by: frgrogt3nreg 27255 |
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