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Theorem 2pthfrgrrn 30044
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 30022 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸))
4 reurex 3374 . . . . . 6 (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
5 prcom 4731 . . . . . . . . . 10 {𝑎, 𝑏} = {𝑏, 𝑎}
65eleq1i 2818 . . . . . . . . 9 ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
76anbi1i 623 . . . . . . . 8 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
8 zfpair2 5421 . . . . . . . . 9 {𝑏, 𝑎} ∈ V
9 zfpair2 5421 . . . . . . . . 9 {𝑏, 𝑐} ∈ V
108, 9prss 4818 . . . . . . . 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 3078 . . . . . 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 3198 . . 3 (𝐺 ∈ USGraph → (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)))
1615imp 406 . 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 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  ∃!wreu 3368  cdif 3940  wss 3943  {csn 4623  {cpr 4625  cfv 6537  Vtxcvtx 28764  Edgcedg 28815  USGraphcusgr 28917   FriendGraph cfrgr 30020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-frgr 30021
This theorem is referenced by:  2pthfrgrrn2  30045  3cyclfrgrrn1  30047
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