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Mirrors > Home > MPE Home > Th. List > frgr2wwlkeu | Structured version Visualization version GIF version |
Description: For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.) |
Ref | Expression |
---|---|
frgr2wwlkeu.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
frgr2wwlkeu | ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1071 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵)) | |
2 | frgr2wwlkeu.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqid 2773 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 2, 3 | frcond2 27817 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
5 | 1, 4 | syl5bir 235 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
6 | 5 | 3impib 1097 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺))) |
7 | frgrusgr 27810 | . . . . . 6 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) | |
8 | usgrumgr 26683 | . . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
9 | 3anan32 1079 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉)) | |
10 | 2, 3 | umgrwwlks2on 27479 | . . . . . . . 8 ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
11 | 10 | ex 405 | . . . . . . 7 ⊢ (𝐺 ∈ UMGraph → ((𝐴 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺))))) |
12 | 9, 11 | syl5bir 235 | . . . . . 6 ⊢ (𝐺 ∈ UMGraph → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) → (〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺))))) |
13 | 7, 8, 12 | 3syl 18 | . . . . 5 ⊢ (𝐺 ∈ FriendGraph → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉) → (〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺))))) |
14 | 13 | impl 448 | . . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
15 | 14 | reubidva 3322 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (∃!𝑐 ∈ 𝑉 〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
16 | 15 | 3adant3 1113 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (∃!𝑐 ∈ 𝑉 〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
17 | 6, 16 | mpbird 249 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 〈“𝐴𝑐𝐵”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∃!wreu 3085 {cpr 4438 ‘cfv 6186 (class class class)co 6975 2c2 11494 〈“cs3 14065 Vtxcvtx 26500 Edgcedg 26551 UMGraphcumgr 26585 USGraphcusgr 26653 WWalksNOn cwwlksnon 27329 FriendGraph cfrgr 27806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-ac2 9682 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-ifp 1045 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-dju 9123 df-card 9161 df-ac 9335 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-xnn0 11779 df-z 11793 df-uz 12058 df-fz 12708 df-fzo 12849 df-hash 13505 df-word 13672 df-concat 13733 df-s1 13758 df-s2 14071 df-s3 14072 df-edg 26552 df-uhgr 26562 df-upgr 26586 df-umgr 26587 df-usgr 26655 df-wlks 27100 df-wwlks 27332 df-wwlksn 27333 df-wwlksnon 27334 df-frgr 27807 |
This theorem is referenced by: frgr2wwlkn0 27878 frgr2wwlk1 27879 frgr2wwlkeqm 27881 |
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