Theorem frgrusgr 29494

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 29493	. 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 3944   ⊆ wss 3947  {csn 4627  {cpr 4629  ‘cfv 6540  Vtxcvtx 28236  Edgcedg 28287  USGraphcusgr 28389   FriendGraph cfrgr 29491

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 5305

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 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-frgr 29492

This theorem is referenced by:  frgreu  29501  frcond3  29502  nfrgr2v  29505  3vfriswmgr  29511  2pthfrgrrn2  29516  2pthfrgr  29517  3cyclfrgrrn2  29520  3cyclfrgr  29521  n4cyclfrgr  29524  frgrnbnb  29526  vdgn0frgrv2  29528  vdgn1frgrv2  29529  frgrncvvdeqlem2  29533  frgrncvvdeqlem3  29534  frgrncvvdeqlem6  29537  frgrncvvdeqlem9  29540  frgrncvvdeq  29542  frgrwopreglem4a  29543  frgrwopreg  29556  frgrregorufrg  29559  frgr2wwlkeu  29560  frgr2wsp1  29563  frgr2wwlkeqm  29564  frrusgrord0lem  29572  frrusgrord0  29573  friendshipgt3  29631