Theorem frcond1 30285

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 30279	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4		preq2 4734	. . . . . . 7 ⊢ (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
5	4	preq1d 4739	. . . . . 6 ⊢ (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
6	5	sseq1d 4015	. . . . 5 ⊢ (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
7	6	reubidv 3398	. . . 4 ⊢ (𝑘 = 𝐴 → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8		preq2 4734	. . . . . . 7 ⊢ (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
9	8	preq2d 4740	. . . . . 6 ⊢ (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
10	9	sseq1d 4015	. . . . 5 ⊢ (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
11	10	reubidv 3398	. . . 4 ⊢ (𝑙 = 𝐶 → (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12		simp1 1137	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → 𝐴 ∈ 𝑉)
13		sneq 4636	. . . . . 6 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
14	13	difeq2d 4126	. . . . 5 ⊢ (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
15	14	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16		necom 2994	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐶 ≠ 𝐴)
17	16	biimpi 216	. . . . . . 7 ⊢ (𝐴 ≠ 𝐶 → 𝐶 ≠ 𝐴)
18	17	anim2i 617	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
19	18	3adant1 1131	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
20		eldifsn 4786	. . . . 5 ⊢ (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐴))
21	19, 20	sylibr 234	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
22	7, 11, 12, 15, 21	rspc2vd 3947	. . 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 4732	. . . . . . 7 ⊢ {𝑏, 𝐴} = {𝐴, 𝑏}
24	23	preq1i 4736	. . . . . 6 ⊢ {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
25	24	sseq1i 4012	. . . . 5 ⊢ ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
26	25	reubii 3389	. . . 4 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
27	26	biimpi 216	. . 3 ⊢ (∃!𝑏 ∈ 𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
28	22, 27	syl6com 37	. 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 504	1 ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃!wreu 3378   ∖ cdif 3948   ⊆ wss 3951  {csn 4626  {cpr 4628  ‘cfv 6561  Vtxcvtx 29013  Edgcedg 29064  USGraphcusgr 29166   FriendGraph cfrgr 30277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-frgr 30278

This theorem is referenced by:  frcond2  30286  frcond3  30288  4cyclusnfrgr  30311  frgrncvvdeqlem2  30319