Theorem frgrusgr 30190

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 2729	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2729	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30189	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 497	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3044  ∃!wreu 3352   ∖ cdif 3911   ⊆ wss 3914  {csn 4589  {cpr 4591  ‘cfv 6511  Vtxcvtx 28923  Edgcedg 28974  USGraphcusgr 29076   FriendGraph cfrgr 30187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-frgr 30188

This theorem is referenced by:  frgreu  30197  frcond3  30198  nfrgr2v  30201  3vfriswmgr  30207  2pthfrgrrn2  30212  2pthfrgr  30213  3cyclfrgrrn2  30216  3cyclfrgr  30217  n4cyclfrgr  30220  frgrnbnb  30222  vdgn0frgrv2  30224  vdgn1frgrv2  30225  frgrncvvdeqlem2  30229  frgrncvvdeqlem3  30230  frgrncvvdeqlem6  30233  frgrncvvdeqlem9  30236  frgrncvvdeq  30238  frgrwopreglem4a  30239  frgrwopreg  30252  frgrregorufrg  30255  frgr2wwlkeu  30256  frgr2wsp1  30259  frgr2wwlkeqm  30260  frrusgrord0lem  30268  frrusgrord0  30269  friendshipgt3  30327