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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 30097 | . . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})) |
4 | eldifsn 4793 | . . . . . 6 ⊢ (𝑙 ∈ (𝑉 ∖ {𝑘}) ↔ (𝑙 ∈ 𝑉 ∧ 𝑙 ≠ 𝑘)) | |
5 | necom 2990 | . . . . . . . 8 ⊢ (𝑙 ≠ 𝑘 ↔ 𝑘 ≠ 𝑙) | |
6 | 5 | biimpi 215 | . . . . . . 7 ⊢ (𝑙 ≠ 𝑘 → 𝑘 ≠ 𝑙) |
7 | 6 | anim2i 615 | . . . . . 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 615 | . . . 4 ⊢ ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → (𝑘 ∈ 𝑉 ∧ (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙))) |
10 | 3anass 1092 | . . . 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 504 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘}))) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}) |
13 | 12 | ralrimivva 3196 | 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 ∖ cdif 3944 ∩ cin 3946 {csn 4630 ‘cfv 6551 (class class class)co 7424 Vtxcvtx 28827 Edgcedg 28878 NeighbVtx cnbgr 29163 FriendGraph cfrgr 30086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-oadd 8495 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-dju 9930 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-n0 12509 df-xnn0 12581 df-z 12595 df-uz 12859 df-fz 13523 df-hash 14328 df-edg 28879 df-upgr 28913 df-umgr 28914 df-usgr 28982 df-nbgr 29164 df-frgr 30087 |
This theorem is referenced by: (None) |
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