Theorem frgrusgr 30241

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 2731	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2731	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30240	. 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 2111  ∀wral 3047  ∃!wreu 3344   ∖ cdif 3894   ⊆ wss 3897  {csn 4573  {cpr 4575  ‘cfv 6481  Vtxcvtx 28974  Edgcedg 29025  USGraphcusgr 29127   FriendGraph cfrgr 30238

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 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-frgr 30239

This theorem is referenced by:  frgreu  30248  frcond3  30249  nfrgr2v  30252  3vfriswmgr  30258  2pthfrgrrn2  30263  2pthfrgr  30264  3cyclfrgrrn2  30267  3cyclfrgr  30268  n4cyclfrgr  30271  frgrnbnb  30273  vdgn0frgrv2  30275  vdgn1frgrv2  30276  frgrncvvdeqlem2  30280  frgrncvvdeqlem3  30281  frgrncvvdeqlem6  30284  frgrncvvdeqlem9  30287  frgrncvvdeq  30289  frgrwopreglem4a  30290  frgrwopreg  30303  frgrregorufrg  30306  frgr2wwlkeu  30307  frgr2wsp1  30310  frgr2wwlkeqm  30311  frrusgrord0lem  30319  frrusgrord0  30320  friendshipgt3  30378