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Theorem 2pthfrgrarn 26274
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.)
Assertion
Ref Expression
2pthfrgrarn (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
Distinct variable groups:   𝐸,𝑎,𝑏,𝑐   𝑉,𝑎,𝑏,𝑐

Proof of Theorem 2pthfrgrarn
StepHypRef Expression
1 frisusgrapr 26256 . 2 (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸))
2 reurex 3041 . . . . . . 7 (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸)
3 prcom 4114 . . . . . . . . . . . 12 {𝑎, 𝑏} = {𝑏, 𝑎}
43eleq1i 2583 . . . . . . . . . . 11 ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑏, 𝑎} ∈ ran 𝐸)
54anbi1i 726 . . . . . . . . . 10 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ↔ ({𝑏, 𝑎} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
6 zfpair2 4733 . . . . . . . . . . 11 {𝑏, 𝑎} ∈ V
7 zfpair2 4733 . . . . . . . . . . 11 {𝑏, 𝑐} ∈ V
86, 7prss 4194 . . . . . . . . . 10 (({𝑏, 𝑎} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸)
95, 8bitri 262 . . . . . . . . 9 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸)
109biimpri 216 . . . . . . . 8 ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
1110reximi 2898 . . . . . . 7 (∃𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
122, 11syl 17 . . . . . 6 (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
1312a1i 11 . . . . 5 (((𝑉 USGrph 𝐸𝑎𝑉) ∧ 𝑐 ∈ (𝑉 ∖ {𝑎})) → (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
1413ralimdva 2849 . . . 4 ((𝑉 USGrph 𝐸𝑎𝑉) → (∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
1514ralimdva 2849 . . 3 (𝑉 USGrph 𝐸 → (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
1615imp 443 . 2 ((𝑉 USGrph 𝐸 ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸) → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
171, 16syl 17 1 (𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1938  wral 2800  wrex 2801  ∃!wreu 2802  cdif 3441  wss 3444  {csn 4028  {cpr 4030   class class class wbr 4481  ran crn 4933   USGrph cusg 25597   FriendGrph cfrgra 26253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-xp 4938  df-rel 4939  df-cnv 4940  df-dm 4942  df-rn 4943  df-frgra 26254
This theorem is referenced by:  2pthfrgrarn2  26275  3cyclfrgrarn1  26277
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