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Theorem frcond1 30191
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 30185 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸))
4 preq2 4742 . . . . . . 7 (𝑘 = 𝐴 → {𝑏, 𝑘} = {𝑏, 𝐴})
54preq1d 4747 . . . . . 6 (𝑘 = 𝐴 → {{𝑏, 𝑘}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝑙}})
65sseq1d 4010 . . . . 5 (𝑘 = 𝐴 → ({{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
76reubidv 3381 . . . 4 (𝑘 = 𝐴 → (∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸))
8 preq2 4742 . . . . . . 7 (𝑙 = 𝐶 → {𝑏, 𝑙} = {𝑏, 𝐶})
98preq2d 4748 . . . . . 6 (𝑙 = 𝐶 → {{𝑏, 𝐴}, {𝑏, 𝑙}} = {{𝑏, 𝐴}, {𝑏, 𝐶}})
109sseq1d 4010 . . . . 5 (𝑙 = 𝐶 → ({{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
1110reubidv 3381 . . . 4 (𝑙 = 𝐶 → (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝑙}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
12 simp1 1133 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐴𝑉)
13 sneq 4642 . . . . . 6 (𝑘 = 𝐴 → {𝑘} = {𝐴})
1413difeq2d 4120 . . . . 5 (𝑘 = 𝐴 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
1514adantl 480 . . . 4 (((𝐴𝑉𝐶𝑉𝐴𝐶) ∧ 𝑘 = 𝐴) → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝐴}))
16 necom 2983 . . . . . . . 8 (𝐴𝐶𝐶𝐴)
1716biimpi 215 . . . . . . 7 (𝐴𝐶𝐶𝐴)
1817anim2i 615 . . . . . 6 ((𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
19183adant1 1127 . . . . 5 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (𝐶𝑉𝐶𝐴))
20 eldifsn 4794 . . . . 5 (𝐶 ∈ (𝑉 ∖ {𝐴}) ↔ (𝐶𝑉𝐶𝐴))
2119, 20sylibr 233 . . . 4 ((𝐴𝑉𝐶𝑉𝐴𝐶) → 𝐶 ∈ (𝑉 ∖ {𝐴}))
227, 11, 12, 15, 21rspc2vd 3942 . . 3 ((𝐴𝑉𝐶𝑉𝐴𝐶) → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑏𝑉 {{𝑏, 𝑘}, {𝑏, 𝑙}} ⊆ 𝐸 → ∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸))
23 prcom 4740 . . . . . . 7 {𝑏, 𝐴} = {𝐴, 𝑏}
2423preq1i 4744 . . . . . 6 {{𝑏, 𝐴}, {𝑏, 𝐶}} = {{𝐴, 𝑏}, {𝑏, 𝐶}}
2524sseq1i 4007 . . . . 5 ({{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2625reubii 3372 . . . 4 (∃!𝑏𝑉 {{𝑏, 𝐴}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2726biimpi 215 . . 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 503 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  ∃!wreu 3361  cdif 3943  wss 3946  {csn 4632  {cpr 4634  cfv 6553  Vtxcvtx 28924  Edgcedg 28975  USGraphcusgr 29077   FriendGraph cfrgr 30183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-frgr 30184
This theorem is referenced by:  frcond2  30192  frcond3  30194  4cyclusnfrgr  30217  frgrncvvdeqlem2  30225
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