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Theorem frcond1 30294
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   𝐸,𝑏   𝐺,𝑏   𝑉,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, 2isfrgr 30288 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4 preq2 4738 . . . . . . 7 (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
54preq1d 4743 . . . . . 6 (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
65sseq1d 4026 . . . . 5 (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
76reubidv 3395 . . . 4 (𝑘 = 𝐴 → (∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8 preq2 4738 . . . . . . 7 (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
98preq2d 4744 . . . . . 6 (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
109sseq1d 4026 . . . . 5 (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
1110reubidv 3395 . . . 4 (𝑙 = 𝐶 → (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12 simp1 1135 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐴𝑉)
13 sneq 4640 . . . . . 6 (𝑘 = 𝐴 → {𝑘} = {𝐴})
1413difeq2d 4135 . . . . 5 (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
1514adantl 481 . . . 4 (((𝐴𝑉𝐶𝑉𝐴𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16 necom 2991 . . . . . . . 8 (𝐴𝐶𝐶𝐴)
1716biimpi 216 . . . . . . 7 (𝐴𝐶𝐶𝐴)
1817anim2i 617 . . . . . 6 ((𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
19183adant1 1129 . . . . 5 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
20 eldifsn 4790 . . . . 5 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉𝐶𝐴))
2119, 20sylibr 234 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
227, 11, 12, 15, 21rspc2vd 3958 . . 3 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
23 prcom 4736 . . . . . . 7 {𝑏, 𝐴} = {𝐴, 𝑏}
2423preq1i 4740 . . . . . 6 {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
2524sseq1i 4023 . . . . 5 ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2625reubii 3386 . . . 4 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2726biimpi 216 . . 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 504 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  ∃!wreu 3375  cdif 3959  wss 3962  {csn 4630  {cpr 4632  cfv 6562  Vtxcvtx 29027  Edgcedg 29078  USGraphcusgr 29180   FriendGraph cfrgr 30286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-frgr 30287
This theorem is referenced by:  frcond2  30295  frcond3  30297  4cyclusnfrgr  30320  frgrncvvdeqlem2  30328
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