Theorem frgrusgr 30741

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 2760	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2760	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30740	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 502	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ∀wral 3076  ∃!wreu 3363   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  {cpr 4586  ‘cfv 6533  Vtxcvtx 29453  Edgcedg 29504  USGraphcusgr 29609   FriendGraph cfrgr 30738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-frgr 30739

This theorem is used by:  frgreu  30748  frcond3  30749  nfrgr2v  30752  3vfriswmgr  30758  2pthfrgrrn2  30763  2pthfrgr  30764  3cyclfrgrrn2  30767  3cyclfrgr  30768  n4cyclfrgr  30771  frgrnbnb  30773  vdgn0frgrv2  30775  vdgn1frgrv2  30776  frgrncvvdeqlem2  30780  frgrncvvdeqlem3  30781  frgrncvvdeqlem6  30784  frgrncvvdeqlem9  30787  frgrncvvdeq  30789  frgrwopreglem4a  30790  frgrwopreg  30803  frgrregorufrg  30806  frgr2wwlkeu  30807  frgr2wsp1  30810  frgr2wwlkeqm  30811  frrusgrord0lem  30819  frrusgrord0  30820  friendshipgt3  30878