Theorem frgrusgr 30285

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 2734	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2734	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30284	. 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 2113  ∀wral 3049  ∃!wreu 3346   ∖ cdif 3896   ⊆ wss 3899  {csn 4578  {cpr 4580  ‘cfv 6490  Vtxcvtx 29018  Edgcedg 29069  USGraphcusgr 29171   FriendGraph cfrgr 30282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-frgr 30283

This theorem is referenced by:  frgreu  30292  frcond3  30293  nfrgr2v  30296  3vfriswmgr  30302  2pthfrgrrn2  30307  2pthfrgr  30308  3cyclfrgrrn2  30311  3cyclfrgr  30312  n4cyclfrgr  30315  frgrnbnb  30317  vdgn0frgrv2  30319  vdgn1frgrv2  30320  frgrncvvdeqlem2  30324  frgrncvvdeqlem3  30325  frgrncvvdeqlem6  30328  frgrncvvdeqlem9  30331  frgrncvvdeq  30333  frgrwopreglem4a  30334  frgrwopreg  30347  frgrregorufrg  30350  frgr2wwlkeu  30351  frgr2wsp1  30354  frgr2wwlkeqm  30355  frrusgrord0lem  30363  frrusgrord0  30364  friendshipgt3  30422