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Theorem 2pthfrgrrn 29268
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.)
Hypotheses
Ref Expression
2pthfrgrrn.v 𝑉 = (Vtx‘𝐺)
2pthfrgrrn.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
2pthfrgrrn (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
Distinct variable groups:   𝐸,𝑎,𝑏,𝑐   𝐺,𝑎,𝑏,𝑐   𝑉,𝑎,𝑏,𝑐

Proof of Theorem 2pthfrgrrn
StepHypRef Expression
1 2pthfrgrrn.v . . 3 𝑉 = (Vtx‘𝐺)
2 2pthfrgrrn.e . . 3 𝐸 = (Edg‘𝐺)
31, 2isfrgr 29246 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸))
4 reurex 3360 . . . . . 6 (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
5 prcom 4698 . . . . . . . . . 10 {𝑎, 𝑏} = {𝑏, 𝑎}
65eleq1i 2829 . . . . . . . . 9 ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
76anbi1i 625 . . . . . . . 8 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
8 zfpair2 5390 . . . . . . . . 9 {𝑏, 𝑎} ∈ V
9 zfpair2 5390 . . . . . . . . 9 {𝑏, 𝑐} ∈ V
108, 9prss 4785 . . . . . . . 8 (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
117, 10sylbbr 235 . . . . . . 7 ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
1211reximi 3088 . . . . . 6 (∃𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
134, 12syl 17 . . . . 5 (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
1413a1i 11 . . . 4 ((𝐺 ∈ USGraph ∧ (𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎}))) → (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)))
1514ralimdvva 3202 . . 3 (𝐺 ∈ USGraph → (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)))
1615imp 408 . 2 ((𝐺 ∈ USGraph ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸) → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
173, 16sylbi 216 1 (𝐺 ∈ FriendGraph → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  ∃!wreu 3354  cdif 3912  wss 3915  {csn 4591  {cpr 4593  cfv 6501  Vtxcvtx 27989  Edgcedg 28040  USGraphcusgr 28142   FriendGraph cfrgr 29244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-frgr 29245
This theorem is referenced by:  2pthfrgrrn2  29269  3cyclfrgrrn1  29271
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