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Theorem frgraun 26523
 Description: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Assertion
Ref Expression
frgraun (𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝑉,𝑏

Proof of Theorem frgraun
StepHypRef Expression
1 frgraunss 26522 . 2 (𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
2 prex 4836 . . . . 5 {𝐴, 𝑏} ∈ V
3 prex 4836 . . . . 5 {𝑏, 𝐶} ∈ V
42, 3prss 4291 . . . 4 (({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)
54bicomi 213 . . 3 ({{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸 ↔ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))
65reubii 3105 . 2 (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸 ↔ ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))
71, 6syl6ib 240 1 (𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ≠ wne 2780  ∃!wreu 2898   ⊆ wss 3540  {cpr 4127   class class class wbr 4583  ran crn 5039   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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  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:  frgrancvvdeqlemC  26566  frgraeu  26581  frg2woteu  26582  numclwwlk2lem1  26629
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