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Theorem frgrusgr 27967
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 2818 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2818 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2isfrgr 27966 . 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 2105  wral 3135  ∃!wreu 3137  cdif 3930  wss 3933  {csn 4557  {cpr 4559  cfv 6348  Vtxcvtx 26708  Edgcedg 26759  USGraphcusgr 26861   FriendGraph cfrgr 27964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-frgr 27965
This theorem is referenced by:  frgreu  27974  frcond3  27975  nfrgr2v  27978  3vfriswmgr  27984  2pthfrgrrn2  27989  2pthfrgr  27990  3cyclfrgrrn2  27993  3cyclfrgr  27994  n4cyclfrgr  27997  frgrnbnb  27999  vdgn0frgrv2  28001  vdgn1frgrv2  28002  frgrncvvdeqlem2  28006  frgrncvvdeqlem3  28007  frgrncvvdeqlem6  28010  frgrncvvdeqlem9  28013  frgrncvvdeq  28015  frgrwopreglem4a  28016  frgrwopreg  28029  frgrregorufrg  28032  frgr2wwlkeu  28033  frgr2wsp1  28036  frgr2wwlkeqm  28037  frrusgrord0lem  28045  frrusgrord0  28046  friendshipgt3  28104
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