Theorem frgrusgr 30683

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 2765	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2765	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30682	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 502	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ∀wral 3081  ∃!wreu 3369   ∖ cdif 3903   ⊆ wss 3906  {csn 4591  {cpr 4593  ‘cfv 6540  Vtxcvtx 29401  Edgcedg 29452  USGraphcusgr 29557   FriendGraph cfrgr 30680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-frgr 30681

This theorem is used by:  frgreu  30690  frcond3  30691  nfrgr2v  30694  3vfriswmgr  30700  2pthfrgrrn2  30705  2pthfrgr  30706  3cyclfrgrrn2  30709  3cyclfrgr  30710  n4cyclfrgr  30713  frgrnbnb  30715  vdgn0frgrv2  30717  vdgn1frgrv2  30718  frgrncvvdeqlem2  30722  frgrncvvdeqlem3  30723  frgrncvvdeqlem6  30726  frgrncvvdeqlem9  30729  frgrncvvdeq  30731  frgrwopreglem4a  30732  frgrwopreg  30745  frgrregorufrg  30748  frgr2wwlkeu  30749  frgr2wsp1  30752  frgr2wwlkeqm  30753  frrusgrord0lem  30761  frrusgrord0  30762  friendshipgt3  30820