Theorem frgrusgr 30871

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 2760	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2760	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30870	. 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 2144  ∀wral 3076  ∃!wreu 3363   ⊆ wss 3896   ∖ cdif 4044  {csn 4584  {cpr 4586  ‘cfv 6543  Vtxcvtx 29583  Edgcedg 29634  USGraphcusgr 29739   FriendGraph cfrgr 30868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-frgr 30869

This theorem is used by:  frgreu  30878  frcond3  30879  nfrgr2v  30882  3vfriswmgr  30888  2pthfrgrrn2  30893  2pthfrgr  30894  3cyclfrgrrn2  30897  3cyclfrgr  30898  n4cyclfrgr  30901  frgrnbnb  30903  vdgn0frgrv2  30905  vdgn1frgrv2  30906  frgrncvvdeqlem2  30910  frgrncvvdeqlem3  30911  frgrncvvdeqlem6  30914  frgrncvvdeqlem9  30917  frgrncvvdeq  30919  frgrwopreglem4a  30920  frgrwopreg  30933  frgrregorufrg  30936  frgr2wwlkeu  30937  frgr2wsp1  30940  frgr2wwlkeqm  30941  frrusgrord0lem  30949  frrusgrord0  30950  friendshipgt3  31008