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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 28218 | . . 3 ⊢ (𝐺 ∈ FriendGraph → ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})) |
4 | eldifsn 4685 | . . . . . 6 ⊢ (𝑙 ∈ (𝑉 ∖ {𝑘}) ↔ (𝑙 ∈ 𝑉 ∧ 𝑙 ≠ 𝑘)) | |
5 | necom 2988 | . . . . . . . 8 ⊢ (𝑙 ≠ 𝑘 ↔ 𝑘 ≠ 𝑙) | |
6 | 5 | biimpi 219 | . . . . . . 7 ⊢ (𝑙 ≠ 𝑘 → 𝑘 ≠ 𝑙) |
7 | 6 | anim2i 620 | . . . . . 6 ⊢ ((𝑙 ∈ 𝑉 ∧ 𝑙 ≠ 𝑘) → (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙)) |
8 | 4, 7 | sylbi 220 | . . . . 5 ⊢ (𝑙 ∈ (𝑉 ∖ {𝑘}) → (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙)) |
9 | 8 | anim2i 620 | . . . 4 ⊢ ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → (𝑘 ∈ 𝑉 ∧ (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙))) |
10 | 3anass 1096 | . . . 4 ⊢ ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙) ↔ (𝑘 ∈ 𝑉 ∧ (𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙))) | |
11 | 9, 10 | sylibr 237 | . . 3 ⊢ ((𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘})) → (𝑘 ∈ 𝑉 ∧ 𝑙 ∈ 𝑉 ∧ 𝑘 ≠ 𝑙)) |
12 | 3, 11 | impel 509 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑘 ∈ 𝑉 ∧ 𝑙 ∈ (𝑉 ∖ {𝑘}))) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}) |
13 | 12 | ralrimivva 3104 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∀wral 3054 ∃wrex 3055 ∖ cdif 3850 ∩ cin 3852 {csn 4526 ‘cfv 6349 (class class class)co 7182 Vtxcvtx 26953 Edgcedg 27004 NeighbVtx cnbgr 27286 FriendGraph cfrgr 28207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-oadd 8147 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-dju 9415 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-n0 11989 df-xnn0 12061 df-z 12075 df-uz 12337 df-fz 12994 df-hash 13795 df-edg 27005 df-upgr 27039 df-umgr 27040 df-usgr 27108 df-nbgr 27287 df-frgr 28208 |
This theorem is referenced by: (None) |
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