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Theorem frgraunss 26522
 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
frgraunss (𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝑉,𝑏

Proof of Theorem frgraunss
Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgrapr 26518 . 2 (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸))
2 sneq 4135 . . . . . . . . . . 11 (𝑎 = 𝐴 → {𝑎} = {𝐴})
32difeq2d 3690 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝐴}))
4 preq2 4213 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → {𝑏, 𝑎} = {𝑏, 𝐴})
54preq1d 4218 . . . . . . . . . . . 12 (𝑎 = 𝐴 → {{𝑏, 𝑎}, {𝑏, 𝑐}} = {{𝑏, 𝐴}, {𝑏, 𝑐}})
65sseq1d 3595 . . . . . . . . . . 11 (𝑎 = 𝐴 → ({{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸))
76reubidv 3103 . . . . . . . . . 10 (𝑎 = 𝐴 → (∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸))
83, 7raleqbidv 3129 . . . . . . . . 9 (𝑎 = 𝐴 → (∀𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 ↔ ∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸))
98rspcva 3280 . . . . . . . 8 ((𝐴𝑉 ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸) → ∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸)
10 elsni 4142 . . . . . . . . . . . . . . 15 (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
1110eqcomd 2616 . . . . . . . . . . . . . 14 (𝐶 ∈ {𝐴} → 𝐴 = 𝐶)
1211necon3ai 2807 . . . . . . . . . . . . 13 (𝐴𝐶 → ¬ 𝐶 ∈ {𝐴})
1312anim2i 591 . . . . . . . . . . . 12 ((𝐶𝑉𝐴𝐶) → (𝐶𝑉 ∧ ¬ 𝐶 ∈ {𝐴}))
14 eldif 3550 . . . . . . . . . . . 12 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉 ∧ ¬ 𝐶 ∈ {𝐴}))
1513, 14sylibr 223 . . . . . . . . . . 11 ((𝐶𝑉𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
16 preq2 4213 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝐶 → {𝑏, 𝑐} = {𝑏, 𝐶})
1716preq2d 4219 . . . . . . . . . . . . . . . 16 (𝑐 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑐}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
1817sseq1d 3595 . . . . . . . . . . . . . . 15 (𝑐 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ ran 𝐸))
1918reubidv 3103 . . . . . . . . . . . . . 14 (𝑐 = 𝐶 → (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ ran 𝐸))
2019rspcva 3280 . . . . . . . . . . . . 13 ((𝐶 ∈ (𝑉 ∖ {𝐴}) ∧ ∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸) → ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ ran 𝐸)
21 prcom 4211 . . . . . . . . . . . . . . . 16 {𝑏, 𝐴} = {𝐴, 𝑏}
2221preq1i 4215 . . . . . . . . . . . . . . 15 {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
2322sseq1i 3592 . . . . . . . . . . . . . 14 ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ ran 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)
2423reubii 3105 . . . . . . . . . . . . 13 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ ran 𝐸 ↔ ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)
2520, 24sylib 207 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑉 ∖ {𝐴}) ∧ ∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)
2625ex 449 . . . . . . . . . . 11 (𝐶 ∈ (𝑉 ∖ {𝐴}) → (∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
2715, 26syl 17 . . . . . . . . . 10 ((𝐶𝑉𝐴𝐶) → (∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
2827ex 449 . . . . . . . . 9 (𝐶𝑉 → (𝐴𝐶 → (∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)))
2928com13 86 . . . . . . . 8 (∀𝑐 ∈ (𝑉 ∖ {𝐴})∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑐}} ⊆ ran 𝐸 → (𝐴𝐶 → (𝐶𝑉 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)))
309, 29syl 17 . . . . . . 7 ((𝐴𝑉 ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸) → (𝐴𝐶 → (𝐶𝑉 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸)))
3130ex 449 . . . . . 6 (𝐴𝑉 → (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → (𝐴𝐶 → (𝐶𝑉 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))))
3231com24 93 . . . . 5 (𝐴𝑉 → (𝐶𝑉 → (𝐴𝐶 → (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))))
33323imp 1249 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
3433com12 32 . . 3 (∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
3534adantl 481 . 2 ((𝑉 USGrph 𝐸 ∧ ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃!𝑏𝑉 {{𝑏, 𝑎}, {𝑏, 𝑐}} ⊆ ran 𝐸) → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
361, 35syl 17 1 (𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃!wreu 2898   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   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:  frgraun  26523  4cyclusnfrgra  26546  frgrancvvdeqlem3  26559
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