Theorem frgrusgr 30293

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 2740	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2740	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30292	. 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 2108  ∀wral 3067  ∃!wreu 3386   ∖ cdif 3973   ⊆ wss 3976  {csn 4648  {cpr 4650  ‘cfv 6573  Vtxcvtx 29031  Edgcedg 29082  USGraphcusgr 29184   FriendGraph cfrgr 30290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-frgr 30291

This theorem is referenced by:  frgreu  30300  frcond3  30301  nfrgr2v  30304  3vfriswmgr  30310  2pthfrgrrn2  30315  2pthfrgr  30316  3cyclfrgrrn2  30319  3cyclfrgr  30320  n4cyclfrgr  30323  frgrnbnb  30325  vdgn0frgrv2  30327  vdgn1frgrv2  30328  frgrncvvdeqlem2  30332  frgrncvvdeqlem3  30333  frgrncvvdeqlem6  30336  frgrncvvdeqlem9  30339  frgrncvvdeq  30341  frgrwopreglem4a  30342  frgrwopreg  30355  frgrregorufrg  30358  frgr2wwlkeu  30359  frgr2wsp1  30362  frgr2wwlkeqm  30363  frrusgrord0lem  30371  frrusgrord0  30372  friendshipgt3  30430