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Mirrors > Home > MPE Home > Th. List > 2pthfrgrrn | Structured version Visualization version GIF version |
Description: Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.) (Revised by AV, 1-Apr-2021.) |
Ref | Expression |
---|---|
2pthfrgrrn.v | ⊢ 𝑉 = (Vtx‘𝐺) |
2pthfrgrrn.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
2pthfrgrrn | ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2pthfrgrrn.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 2pthfrgrrn.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | 1, 2 | frgrusgrfrcond 27730 | . 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)) |
4 | reurex 3391 | . . . . . 6 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸) | |
5 | prcom 4575 | . . . . . . . . . 10 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎} | |
6 | 5 | eleq1i 2873 | . . . . . . . . 9 ⊢ ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸) |
7 | 6 | anbi1i 623 | . . . . . . . 8 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
8 | zfpair2 5222 | . . . . . . . . 9 ⊢ {𝑏, 𝑎} ∈ V | |
9 | zfpair2 5222 | . . . . . . . . 9 ⊢ {𝑏, 𝑐} ∈ V | |
10 | 8, 9 | prss 4660 | . . . . . . . 8 ⊢ (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸) |
11 | 7, 10 | sylbbr 237 | . . . . . . 7 ⊢ ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
12 | 11 | reximi 3207 | . . . . . 6 ⊢ (∃𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
13 | 4, 12 | syl 17 | . . . . 5 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
14 | 13 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ (𝑉 ∖ {𝑎}))) → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))) |
15 | 14 | ralimdvva 3146 | . . 3 ⊢ (𝐺 ∈ USGraph → (∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))) |
16 | 15 | imp 407 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸) → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
17 | 3, 16 | sylbi 218 | 1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 ∃wrex 3106 ∃!wreu 3107 ∖ cdif 3856 ⊆ wss 3859 {csn 4472 {cpr 4474 ‘cfv 6225 Vtxcvtx 26464 Edgcedg 26515 USGraphcusgr 26617 FriendGraph cfrgr 27727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-iota 6189 df-fv 6233 df-frgr 27728 |
This theorem is referenced by: 2pthfrgrrn2 27754 3cyclfrgrrn1 27756 |
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