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Mirrors > Home > MPE Home > Th. List > frcond4 | Structured version Visualization version GIF version |
Description: The friendship condition, alternatively expressed by neighborhoods: in a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) (Proof shortened by AV, 30-Dec-2021.) |
Ref | Expression |
---|---|
frcond1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
frcond1.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
frcond4 | ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frcond1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frcond1.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | frcond3 30078 | . . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})) |
4 | eldifsn 4791 | . . . . . 6 ⊢ (𝑙 ∈ (𝑉 ∖ {𝑘}) ↔ (𝑙 ∈ 𝑉 ∧ 𝑙 ≠ 𝑘)) | |
5 | necom 2991 | . . . . . . . 8 ⊢ (𝑙 ≠ 𝑘 ↔ 𝑘 ≠ 𝑙) | |
6 | 5 | biimpi 215 | . . . . . . 7 ⊢ (𝑙 ≠ 𝑘 → 𝑘 ≠ 𝑙) |
7 | 6 | anim2i 616 | . . . . . 6 ⊢ ((𝑙 ∈ 𝑉 ∧ 𝑙 ≠ 𝑘) → (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙)) |
8 | 4, 7 | sylbi 216 | . . . . 5 ⊢ (𝑙 ∈ (𝑉 ∖ {𝑘}) → (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙)) |
9 | 8 | anim2i 616 | . . . 4 ⊢ ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → (𝑘 ∈ 𝑉 ∧ (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙))) |
10 | 3anass 1093 | . . . 4 ⊢ ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙) ↔ (𝑘 ∈ 𝑉 ∧ (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙))) | |
11 | 9, 10 | sylibr 233 | . . 3 ⊢ ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → (𝑘 ∈ 𝑉 ∧ 𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙)) |
12 | 3, 11 | impel 505 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘}))) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}) |
13 | 12 | ralrimivva 3197 | 1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∖ cdif 3944 ∩ cin 3946 {csn 4629 ‘cfv 6548 (class class class)co 7420 Vtxcvtx 28808 Edgcedg 28859 NeighbVtx cnbgr 29144 FriendGraph cfrgr 30067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-oadd 8490 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-hash 14322 df-edg 28860 df-upgr 28894 df-umgr 28895 df-usgr 28963 df-nbgr 29145 df-frgr 30068 |
This theorem is referenced by: (None) |
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