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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 1090 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵)) | |
2 | frgr2wwlkeu.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqid 2737 | . . . . 5 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 2, 3 | frcond2 29253 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
5 | 1, 4 | biimtrrid 242 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺)))) |
6 | 5 | 3impib 1117 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ({𝐴, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝐵} ∈ (Edg‘𝐺))) |
7 | frgrusgr 29247 | . . . . . 6 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) | |
8 | usgrumgr 28172 | . . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
9 | 3anan32 1098 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ↔ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝑐 ∈ 𝑉)) | |
10 | 2, 3 | umgrwwlks2on 28944 | . . . . . . . 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 414 | . . . . . . 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 | biimtrrid 242 | . . . . . 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 457 | . . . 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 3372 | . . 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 1133 | . 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 257 | 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 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∃!wreu 3354 {cpr 4593 ‘cfv 6501 (class class class)co 7362 2c2 12215 ⟨“cs3 14738 Vtxcvtx 27989 Edgcedg 28040 UMGraphcumgr 28074 USGraphcusgr 28142 WWalksNOn cwwlksnon 28814 FriendGraph cfrgr 29244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-ac2 10406 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-ac 10059 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-concat 14466 df-s1 14491 df-s2 14744 df-s3 14745 df-edg 28041 df-uhgr 28051 df-upgr 28075 df-umgr 28076 df-usgr 28144 df-wlks 28589 df-wwlks 28817 df-wwlksn 28818 df-wwlksnon 28819 df-frgr 29245 |
This theorem is referenced by: frgr2wwlkn0 29314 frgr2wwlk1 29315 frgr2wwlkeqm 29317 |
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