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Theorem frgrusgr 28625
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 2738 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2738 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2isfrgr 28624 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
43simplbi 498 1 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3064  ∃!wreu 3066  cdif 3884  wss 3887  {csn 4561  {cpr 4563  cfv 6433  Vtxcvtx 27366  Edgcedg 27417  USGraphcusgr 27519   FriendGraph cfrgr 28622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-frgr 28623
This theorem is referenced by:  frgreu  28632  frcond3  28633  nfrgr2v  28636  3vfriswmgr  28642  2pthfrgrrn2  28647  2pthfrgr  28648  3cyclfrgrrn2  28651  3cyclfrgr  28652  n4cyclfrgr  28655  frgrnbnb  28657  vdgn0frgrv2  28659  vdgn1frgrv2  28660  frgrncvvdeqlem2  28664  frgrncvvdeqlem3  28665  frgrncvvdeqlem6  28668  frgrncvvdeqlem9  28671  frgrncvvdeq  28673  frgrwopreglem4a  28674  frgrwopreg  28687  frgrregorufrg  28690  frgr2wwlkeu  28691  frgr2wsp1  28694  frgr2wwlkeqm  28695  frrusgrord0lem  28703  frrusgrord0  28704  friendshipgt3  28762
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