Theorem frgrusgr 30188

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 2735	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2735	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30187	. 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 3051  ∃!wreu 3357   ∖ cdif 3923   ⊆ wss 3926  {csn 4601  {cpr 4603  ‘cfv 6530  Vtxcvtx 28921  Edgcedg 28972  USGraphcusgr 29074   FriendGraph cfrgr 30185

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-frgr 30186

This theorem is referenced by:  frgreu  30195  frcond3  30196  nfrgr2v  30199  3vfriswmgr  30205  2pthfrgrrn2  30210  2pthfrgr  30211  3cyclfrgrrn2  30214  3cyclfrgr  30215  n4cyclfrgr  30218  frgrnbnb  30220  vdgn0frgrv2  30222  vdgn1frgrv2  30223  frgrncvvdeqlem2  30227  frgrncvvdeqlem3  30228  frgrncvvdeqlem6  30231  frgrncvvdeqlem9  30234  frgrncvvdeq  30236  frgrwopreglem4a  30237  frgrwopreg  30250  frgrregorufrg  30253  frgr2wwlkeu  30254  frgr2wsp1  30257  frgr2wwlkeqm  30258  frrusgrord0lem  30266  frrusgrord0  30267  friendshipgt3  30325