Theorem frgrusgr 30349

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 2739	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2739	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30348	. 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 2119  ∀wral 3053  ∃!wreu 3342   ∖ cdif 3880   ⊆ wss 3883  {csn 4555  {cpr 4557  ‘cfv 6485  Vtxcvtx 29083  Edgcedg 29134  USGraphcusgr 29236   FriendGraph cfrgr 30346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-frgr 30347

This theorem is referenced by:  frgreu  30356  frcond3  30357  nfrgr2v  30360  3vfriswmgr  30366  2pthfrgrrn2  30371  2pthfrgr  30372  3cyclfrgrrn2  30375  3cyclfrgr  30376  n4cyclfrgr  30379  frgrnbnb  30381  vdgn0frgrv2  30383  vdgn1frgrv2  30384  frgrncvvdeqlem2  30388  frgrncvvdeqlem3  30389  frgrncvvdeqlem6  30392  frgrncvvdeqlem9  30395  frgrncvvdeq  30397  frgrwopreglem4a  30398  frgrwopreg  30411  frgrregorufrg  30414  frgr2wwlkeu  30415  frgr2wsp1  30418  frgr2wwlkeqm  30419  frrusgrord0lem  30427  frrusgrord0  30428  friendshipgt3  30486