Theorem 2pthfrgrarn 26536

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 → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))

Distinct variable groups:   𝐸,𝑎,𝑏,𝑐   𝑉,𝑎,𝑏,𝑐

Proof of Theorem 2pthfrgrarn

Step	Hyp	Ref	Expression
1		frisusgrapr 26518	. 2 ⊢ (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸))
2		reurex 3137	. . . . . . 7 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸)
3		prcom 4211	. . . . . . . . . . . 12 ⊢ {𝑎, 𝑏} = {𝑏, 𝑎}
4	3	eleq1i 2679	. . . . . . . . . . 11 ⊢ ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑏, 𝑎} ∈ ran 𝐸)
5	4	anbi1i 727	. . . . . . . . . 10 ⊢ (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ↔ ({𝑏, 𝑎} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
6		zfpair2 4834	. . . . . . . . . . 11 ⊢ {𝑏, 𝑎} ∈ V
7		zfpair2 4834	. . . . . . . . . . 11 ⊢ {𝑏, 𝑐} ∈ V
8	6, 7	prss 4291	. . . . . . . . . 10 ⊢ (({𝑏, 𝑎} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸)
9	5, 8	bitri 263	. . . . . . . . 9 ⊢ (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ↔ {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸)
10	9	biimpri 217	. . . . . . . 8 ⊢ ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
11	10	reximi 2994	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
12	2, 11	syl 17	. . . . . 6 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
13	12	a1i 11	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑎 ∈ 𝑉) ∧ 𝑐 ∈ (𝑉 ∖ {𝑎})) → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
14	13	ralimdva 2945	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑎 ∈ 𝑉) → (∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
15	14	ralimdva 2945	. . 3 ⊢ (𝑉 USGrph 𝐸 → (∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸)))
16	15	imp 444	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸) → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
17	1, 16	syl 17	1 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   FriendGrph cfrgra 26515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-frgra 26516

This theorem is referenced by:  2pthfrgrarn2  26537  3cyclfrgrarn1  26539