Theorem frgrusgr 29514

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 2733	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2733	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 29513	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 499	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3062  ∃!wreu 3375   ∖ cdif 3946   ⊆ wss 3949  {csn 4629  {cpr 4631  ‘cfv 6544  Vtxcvtx 28256  Edgcedg 28307  USGraphcusgr 28409   FriendGraph cfrgr 29511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-frgr 29512

This theorem is referenced by:  frgreu  29521  frcond3  29522  nfrgr2v  29525  3vfriswmgr  29531  2pthfrgrrn2  29536  2pthfrgr  29537  3cyclfrgrrn2  29540  3cyclfrgr  29541  n4cyclfrgr  29544  frgrnbnb  29546  vdgn0frgrv2  29548  vdgn1frgrv2  29549  frgrncvvdeqlem2  29553  frgrncvvdeqlem3  29554  frgrncvvdeqlem6  29557  frgrncvvdeqlem9  29560  frgrncvvdeq  29562  frgrwopreglem4a  29563  frgrwopreg  29576  frgrregorufrg  29579  frgr2wwlkeu  29580  frgr2wsp1  29583  frgr2wwlkeqm  29584  frrusgrord0lem  29592  frrusgrord0  29593  friendshipgt3  29651