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Theorem frgrusgr 29208
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 2737 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2737 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2isfrgr 29207 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
43simplbi 499 1 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3065  ∃!wreu 3352  cdif 3908  wss 3911  {csn 4587  {cpr 4589  cfv 6497  Vtxcvtx 27950  Edgcedg 28001  USGraphcusgr 28103   FriendGraph cfrgr 29205
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-frgr 29206
This theorem is referenced by:  frgreu  29215  frcond3  29216  nfrgr2v  29219  3vfriswmgr  29225  2pthfrgrrn2  29230  2pthfrgr  29231  3cyclfrgrrn2  29234  3cyclfrgr  29235  n4cyclfrgr  29238  frgrnbnb  29240  vdgn0frgrv2  29242  vdgn1frgrv2  29243  frgrncvvdeqlem2  29247  frgrncvvdeqlem3  29248  frgrncvvdeqlem6  29251  frgrncvvdeqlem9  29254  frgrncvvdeq  29256  frgrwopreglem4a  29257  frgrwopreg  29270  frgrregorufrg  29273  frgr2wwlkeu  29274  frgr2wsp1  29277  frgr2wwlkeqm  29278  frrusgrord0lem  29286  frrusgrord0  29287  friendshipgt3  29345
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