Theorem frgrusgr 30552

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.

Step	Hyp	Ref	Expression
1		eqid 2769	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2769	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30551	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 501	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2149  ∀wral 3085  ∃!wreu 3374   ∖ cdif 3910   ⊆ wss 3913  {csn 4594  {cpr 4596  ‘cfv 6537  Vtxcvtx 29286  Edgcedg 29337  USGraphcusgr 29439   FriendGraph cfrgr 30549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-frgr 30550

This theorem is referenced by:  frgreu  30559  frcond3  30560  nfrgr2v  30563  3vfriswmgr  30569  2pthfrgrrn2  30574  2pthfrgr  30575  3cyclfrgrrn2  30578  3cyclfrgr  30579  n4cyclfrgr  30582  frgrnbnb  30584  vdgn0frgrv2  30586  vdgn1frgrv2  30587  frgrncvvdeqlem2  30591  frgrncvvdeqlem3  30592  frgrncvvdeqlem6  30595  frgrncvvdeqlem9  30598  frgrncvvdeq  30600  frgrwopreglem4a  30601  frgrwopreg  30614  frgrregorufrg  30617  frgr2wwlkeu  30618  frgr2wsp1  30621  frgr2wwlkeqm  30622  frrusgrord0lem  30630  frrusgrord0  30631  friendshipgt3  30689