Theorem frgrusgr 27240

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 2651	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2651	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	frgrusgrfrcond 27239	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 475	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2030  ∀wral 2941  ∃!wreu 2943   ∖ cdif 3604   ⊆ wss 3607  {csn 4210  {cpr 4212  ‘cfv 5926  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   FriendGraph cfrgr 27236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-frgr 27237

This theorem is referenced by:  frgreu  27248  frcond3  27249  nfrgr2v  27252  3vfriswmgr  27258  2pthfrgrrn2  27263  2pthfrgr  27264  3cyclfrgrrn2  27267  3cyclfrgr  27268  n4cyclfrgr  27271  frgrnbnb  27273  vdgn0frgrv2  27275  vdgn1frgrv2  27276  frgrncvvdeqlem2  27280  frgrncvvdeqlem3  27281  frgrncvvdeqlem6  27284  frgrncvvdeqlem9  27287  frgrncvvdeq  27289  frgrwopreglem4a  27290  frgrwopreg  27303  frgrregorufrg  27306  frgr2wwlkeu  27307  frgr2wsp1  27310  frgr2wwlkeqm  27311  frrusgrord0lem  27319  frrusgrord0  27320  friendshipgt3  27385