Theorem 2pthfrgrrn 27146

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:   𝐺,𝑎,𝑏,𝑐   𝑉,𝑎,𝑏,𝑐

Allowed substitution hints:   𝐸(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	frgrusgrfrcond 27123	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸))
4		reurex 3160	. . . . . 6 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ∃𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
5		prcom 4267	. . . . . . . . . 10 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
6	5	eleq1i 2692	. . . . . . . . 9 ⊢ ({𝑎, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝑎} ∈ 𝐸)
7	6	anbi1i 731	. . . . . . . 8 ⊢ (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ ({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
8		zfpair2 4907	. . . . . . . . 9 ⊢ {𝑏, 𝑎} ∈ V
9		zfpair2 4907	. . . . . . . . 9 ⊢ {𝑏, 𝑐} ∈ V
10	8, 9	prss 4351	. . . . . . . 8 ⊢ (({𝑏, 𝑎} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸)
11	7, 10	sylbbr 226	. . . . . . 7 ⊢ ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ 𝐸 → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))
12	11	reximi 3011	. . . . . 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 2964	. . 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 445	. 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 207	1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  {cpr 4179  ‘cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-frgr 27121

This theorem is referenced by:  2pthfrgrrn2  27147  3cyclfrgrrn1  27149