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Mirrors > Home > MPE Home > Th. List > 3cyclfrgrrn | Structured version Visualization version GIF version |
Description: Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.) (Revised by AV, 2-Apr-2021.) |
Ref | Expression |
---|---|
3cyclfrgrrn1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
3cyclfrgrrn1.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
3cyclfrgrrn | ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3cyclfrgrrn1.v | . . . . . . . . 9 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | fvexi 6920 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
3 | hashgt12el2 14458 | . . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 1 < (♯‘𝑉) ∧ 𝑎 ∈ 𝑉) → ∃𝑥 ∈ 𝑉 𝑎 ≠ 𝑥) | |
4 | 2, 3 | mp3an1 1447 | . . . . . . 7 ⊢ ((1 < (♯‘𝑉) ∧ 𝑎 ∈ 𝑉) → ∃𝑥 ∈ 𝑉 𝑎 ≠ 𝑥) |
5 | simpr 484 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥 ∧ 𝑎 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝐺 ∈ FriendGraph ) | |
6 | pm3.22 459 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) → (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) | |
7 | 6 | 3adant2 1130 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥 ∧ 𝑎 ∈ 𝑉) → (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) |
8 | 7 | adantr 480 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥 ∧ 𝑎 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) |
9 | simpl2 1191 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥 ∧ 𝑎 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → 𝑎 ≠ 𝑥) | |
10 | 3cyclfrgrrn1.e | . . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺) | |
11 | 1, 10 | 3cyclfrgrrn1 30313 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉) ∧ 𝑎 ≠ 𝑥) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) |
12 | 5, 8, 9, 11 | syl3anc 1370 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥 ∧ 𝑎 ∈ 𝑉) ∧ 𝐺 ∈ FriendGraph ) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) |
13 | 12 | 3exp1 1351 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑉 → (𝑎 ≠ 𝑥 → (𝑎 ∈ 𝑉 → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))))) |
14 | 13 | rexlimiv 3145 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝑉 𝑎 ≠ 𝑥 → (𝑎 ∈ 𝑉 → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))) |
15 | 4, 14 | syl 17 | . . . . . 6 ⊢ ((1 < (♯‘𝑉) ∧ 𝑎 ∈ 𝑉) → (𝑎 ∈ 𝑉 → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))) |
16 | 15 | expcom 413 | . . . . 5 ⊢ (𝑎 ∈ 𝑉 → (1 < (♯‘𝑉) → (𝑎 ∈ 𝑉 → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))))) |
17 | 16 | pm2.43a 54 | . . . 4 ⊢ (𝑎 ∈ 𝑉 → (1 < (♯‘𝑉) → (𝐺 ∈ FriendGraph → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))) |
18 | 17 | com13 88 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (1 < (♯‘𝑉) → (𝑎 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))) |
19 | 18 | imp 406 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → (𝑎 ∈ 𝑉 → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))) |
20 | 19 | ralrimiv 3142 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 Vcvv 3477 {cpr 4632 class class class wbr 5147 ‘cfv 6562 1c1 11153 < clt 11292 ♯chash 14365 Vtxcvtx 29027 Edgcedg 29078 FriendGraph cfrgr 30286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366 df-edg 29079 df-umgr 29114 df-usgr 29182 df-frgr 30287 |
This theorem is referenced by: 3cyclfrgrrn2 30315 3cyclfrgr 30316 |
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