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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 | isfrgr 28041 | . 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)) |
4 | reurex 3433 | . . . . . 6 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸) | |
5 | prcom 4670 | . . . . . . . . . 10 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎} | |
6 | 5 | eleq1i 2905 | . . . . . . . . 9 ⊢ ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸) |
7 | 6 | anbi1i 625 | . . . . . . . 8 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
8 | zfpair2 5333 | . . . . . . . . 9 ⊢ {𝑏, 𝑎} ∈ V | |
9 | zfpair2 5333 | . . . . . . . . 9 ⊢ {𝑏, 𝑐} ∈ V | |
10 | 8, 9 | prss 4755 | . . . . . . . 8 ⊢ (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸) |
11 | 7, 10 | sylbbr 238 | . . . . . . 7 ⊢ ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) |
12 | 11 | reximi 3245 | . . . . . 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 3181 | . . 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 409 | . 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 219 | 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∃!wreu 3142 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 {cpr 4571 ‘cfv 6357 Vtxcvtx 26783 Edgcedg 26834 USGraphcusgr 26936 FriendGraph cfrgr 28039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-frgr 28040 |
This theorem is referenced by: 2pthfrgrrn2 28064 3cyclfrgrrn1 28066 |
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