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Theorem frgrusgr 41432
 Description: A friendship graph is a simple graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
Assertion
Ref Expression
frgrusgr (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )

Proof of Theorem frgrusgr
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2610 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2frgrusgrfrcond 41431 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
43simplbi 475 1 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  ‘cfv 5804  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   FriendGraph cfrgr 41428 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 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-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-sbc 3403  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-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-frgr 41429 This theorem is referenced by:  nfrgr2v  41442  3vfriswmgr  41448  2pthfrgrrn2  41453  2pthfrgr  41454  3cyclfrgrrn2  41457  3cyclfrgr  41458  n4cyclfrgr  41461  frgrnbnb  41463  vdgn0frgrv2  41465  vdgn1frgrv2  41466  frgrncvvdeqlem2  41470  frgrncvvdeqlem3  41471  frgrncvvdeqlem4  41472  frgrncvvdeqlem7  41475  frgrncvvdeqlemC  41478  frgrncvvdeq  41480  frgrwopreglem4  41484  frgrwopreg  41486  frgreu  41491  frgr2wwlkeu  41492  frgr2wsp1  41495  frgr2wwlkeqm  41496  frrusgrord0  41503  frgrregorufrg  41505  av-friendshipgt3  41552
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