Theorem frgrusgr 30919

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 2763	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2763	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30918	. 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 2147  ∀wral 3080  ∃!wreu 3367   ⊆ wss 3900   ∖ cdif 4048  {csn 4588  {cpr 4590  ‘cfv 6566  Vtxcvtx 29631  Edgcedg 29682  USGraphcusgr 29787   FriendGraph cfrgr 30916

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-frgr 30917

This theorem is used by:  frgreu  30926  frcond3  30927  nfrgr2v  30930  3vfriswmgr  30936  2pthfrgrrn2  30941  2pthfrgr  30942  3cyclfrgrrn2  30945  3cyclfrgr  30946  n4cyclfrgr  30949  frgrnbnb  30951  vdgn0frgrv2  30953  vdgn1frgrv2  30954  frgrncvvdeqlem2  30958  frgrncvvdeqlem3  30959  frgrncvvdeqlem6  30962  frgrncvvdeqlem9  30965  frgrncvvdeq  30967  frgrwopreglem4a  30968  frgrwopreg  30981  frgrregorufrg  30984  frgr2wwlkeu  30985  frgr2wsp1  30988  frgr2wwlkeqm  30989  frrusgrord0lem  30997  frrusgrord0  30998  friendshipgt3  31056