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Theorem frcond1 27420
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 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐺,𝑏   𝑉,𝑏
Allowed substitution hint:   𝐸(𝑏)

Proof of Theorem frcond1
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frcond1.v . . 3 𝑉 = (Vtx‘𝐺)
2 frcond1.e . . 3 𝐸 = (Edg‘𝐺)
31, 2frgrusgrfrcond 27413 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4 preq2 4413 . . . . . . 7 (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
54preq1d 4418 . . . . . 6 (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
65sseq1d 3773 . . . . 5 (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
76reubidv 3265 . . . 4 (𝑘 = 𝐴 → (∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8 preq2 4413 . . . . . . 7 (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
98preq2d 4419 . . . . . 6 (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
109sseq1d 3773 . . . . 5 (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
1110reubidv 3265 . . . 4 (𝑙 = 𝐶 → (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12 simp1 1131 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐴𝑉)
13 sneq 4331 . . . . . 6 (𝑘 = 𝐴 → {𝑘} = {𝐴})
1413difeq2d 3871 . . . . 5 (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
1514adantl 473 . . . 4 (((𝐴𝑉𝐶𝑉𝐴𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16 necom 2985 . . . . . . . 8 (𝐴𝐶𝐶𝐴)
1716biimpi 206 . . . . . . 7 (𝐴𝐶𝐶𝐴)
1817anim2i 594 . . . . . 6 ((𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
19183adant1 1125 . . . . 5 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
20 eldifsn 4462 . . . . 5 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉𝐶𝐴))
2119, 20sylibr 224 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
227, 11, 12, 15, 21rspc2vd 27419 . . 3 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
23 prcom 4411 . . . . . . 7 {𝑏, 𝐴} = {𝐴, 𝑏}
2423preq1i 4415 . . . . . 6 {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
2524sseq1i 3770 . . . . 5 ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2625reubii 3267 . . . 4 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2726biimpi 206 . . 3 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2822, 27syl6com 37 . 2 (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
293, 28simplbiim 661 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  ∃!wreu 3052  cdif 3712  wss 3715  {csn 4321  {cpr 4323  cfv 6049  Vtxcvtx 26073  Edgcedg 26138  USGraphcusgr 26243   FriendGraph cfrgr 27410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-frgr 27411
This theorem is referenced by:  frcond2  27421  frcond3  27423  4cyclusnfrgr  27446  frgrncvvdeqlem2  27454
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