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Theorem frisusgra 26519
 Description: A friendship graph is an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frisusgra (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)

Proof of Theorem frisusgra
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgrapr 26518 . 2 (𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
21simpld 474 1 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀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-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:  frgra0v  26520  frgra2v  26526  3vfriswmgra  26532  2pthfrgrarn2  26537  2pthfrgra  26538  3cyclfrgrarn  26540  3cyclfrgrarn2  26541  3cyclfrgra  26542  n4cyclfrgra  26545  frgranbnb  26547  frconngra  26548  vdfrgra0  26549  vdn0frgrav2  26550  vdgn0frgrav2  26551  vdn1frgrav2  26552  vdgn1frgrav2  26553  vdgfrgragt2  26554  frgrancvvdeqlem2  26558  frgrancvvdeqlem3  26559  frgrancvvdeqlem4  26560  frgrancvvdeqlem7  26563  frgrancvvdeqlemC  26566  frgrancvvdeq  26569  frgrancvvdgeq  26570  frgrawopreglem1  26571  frgrawopreglem2  26572  frgrawopreglem4  26574  frgrawopreg  26576  frgraeu  26581  frg2woteu  26582  frg2wot1  26584  frg2spot1  26585  frg2woteqm  26586  frghash2spot  26590  frgregordn0  26597  frgraregorufrg  26599  frgrareggt1  26643  friendshipgt3  26648
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