MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgrusgr Structured version   Visualization version   GIF version

Theorem frgrusgr 28032
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 2819 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2819 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2isfrgr 28031 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
43simplbi 500 1 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wral 3136  ∃!wreu 3138  cdif 3931  wss 3934  {csn 4559  {cpr 4561  cfv 6348  Vtxcvtx 26773  Edgcedg 26824  USGraphcusgr 26926   FriendGraph cfrgr 28029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-frgr 28030
This theorem is referenced by:  frgreu  28039  frcond3  28040  nfrgr2v  28043  3vfriswmgr  28049  2pthfrgrrn2  28054  2pthfrgr  28055  3cyclfrgrrn2  28058  3cyclfrgr  28059  n4cyclfrgr  28062  frgrnbnb  28064  vdgn0frgrv2  28066  vdgn1frgrv2  28067  frgrncvvdeqlem2  28071  frgrncvvdeqlem3  28072  frgrncvvdeqlem6  28075  frgrncvvdeqlem9  28078  frgrncvvdeq  28080  frgrwopreglem4a  28081  frgrwopreg  28094  frgrregorufrg  28097  frgr2wwlkeu  28098  frgr2wsp1  28101  frgr2wwlkeqm  28102  frrusgrord0lem  28110  frrusgrord0  28111  friendshipgt3  28169
  Copyright terms: Public domain W3C validator