Theorem 2pthfrgrrn 28547

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 → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({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

Step	Hyp	Ref	Expression
1		2pthfrgrrn.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		2pthfrgrrn.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	isfrgr 28525	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸))
4		reurex 3352	. . . . . 6 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
5		prcom 4665	. . . . . . . . . 10 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
6	5	eleq1i 2829	. . . . . . . . 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 5348	. . . . . . . . 9 ⊢ {𝑏, 𝑎} ∈ V
9		zfpair2 5348	. . . . . . . . 9 ⊢ {𝑏, 𝑐} ∈ V
10	8, 9	prss 4750	. . . . . . . 8 ⊢ (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
11	7, 10	sylbbr 235	. . . . . . 7 ⊢ ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
12	11	reximi 3174	. . . . . 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 3104	. . 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 406	. 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 216	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065   ∖ cdif 3880   ⊆ wss 3883  {csn 4558  {cpr 4560  ‘cfv 6418  Vtxcvtx 27269  Edgcedg 27320  USGraphcusgr 27422   FriendGraph cfrgr 28523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-frgr 28524

This theorem is referenced by:  2pthfrgrrn2  28548  3cyclfrgrrn1  28550