Theorem frcond1 30560

Description: The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
frcond1.v	⊢ 𝑉 = (Vtx‘𝐺)
frcond1.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
frcond1	⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))

Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝐺,𝑏   𝑉,𝑏

Proof of Theorem frcond1

Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frcond1.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		frcond1.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	isfrgr 30554	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4		preq2 4705	. . . . . . 7 ⊢ (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
5	4	preq1d 4710	. . . . . 6 ⊢ (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
6	5	sseq1d 3976	. . . . 5 ⊢ (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
7	6	reubidv 3392	. . . 4 ⊢ (𝑘 = 𝐴 → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8		preq2 4705	. . . . . . 7 ⊢ (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
9	8	preq2d 4711	. . . . . 6 ⊢ (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
10	9	sseq1d 3976	. . . . 5 ⊢ (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
11	10	reubidv 3392	. . . 4 ⊢ (𝑙 = 𝐶 → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12		simp1 1152	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ 𝑉)
13		sneq 4604	. . . . . 6 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
14	13	difeq2d 4089	. . . . 5 ⊢ (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
15	14	adantl 486	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16		necom 3017	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
17	16	biimpi 219	. . . . . . 7 ⊢ (𝐴 ≠ 𝐶 → 𝐶 ≠ 𝐴)
18	17	anim2i 628	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
19	18	3adant1 1146	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
20		eldifsn 4758	. . . . 5 ⊢ (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
21	19, 20	sylibr 237	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
22	7, 11, 12, 15, 21	rspc2vd 3909	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
23		prcom 4703	. . . . . . 7 ⊢ {𝑏, 𝐴} = {𝐴, 𝑏}
24	23	preq1i 4707	. . . . . 6 ⊢ {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
25	24	sseq1i 3973	. . . . 5 ⊢ ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
26	25	reubii 3385	. . . 4 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
27	26	biimpi 219	. . 3 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
28	22, 27	syl6com 38	. 2 ⊢ (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
29	3, 28	simplbiim 513	1 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃!wreu 3374   ∖ cdif 3910   ⊆ wss 3913  {csn 4594  {cpr 4596  ‘cfv 6539  Vtxcvtx 29289  Edgcedg 29340  USGraphcusgr 29442   FriendGraph cfrgr 30552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-nul 5273

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6495  df-fv 6547  df-frgr 30553

This theorem is referenced by:  frcond2  30561  frcond3  30563  4cyclusnfrgr  30586  frgrncvvdeqlem2  30594