Theorem frgrusgr 30855

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 2761	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2761	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30854	. 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 3077  ∃!wreu 3364   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  {cpr 4586  ‘cfv 6537  Vtxcvtx 29567  Edgcedg 29618  USGraphcusgr 29723   FriendGraph cfrgr 30852

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 2733  ax-nul 5260

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 6493  df-fv 6545  df-frgr 30853

This theorem is used by:  frgreu  30862  frcond3  30863  nfrgr2v  30866  3vfriswmgr  30872  2pthfrgrrn2  30877  2pthfrgr  30878  3cyclfrgrrn2  30881  3cyclfrgr  30882  n4cyclfrgr  30885  frgrnbnb  30887  vdgn0frgrv2  30889  vdgn1frgrv2  30890  frgrncvvdeqlem2  30894  frgrncvvdeqlem3  30895  frgrncvvdeqlem6  30898  frgrncvvdeqlem9  30901  frgrncvvdeq  30903  frgrwopreglem4a  30904  frgrwopreg  30917  frgrregorufrg  30920  frgr2wwlkeu  30921  frgr2wsp1  30924  frgr2wwlkeqm  30925  frrusgrord0lem  30933  frrusgrord0  30934  friendshipgt3  30992