Theorem frgrusgr 30336

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 2736	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2736	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 30335	. 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 3051  ∃!wreu 3348   ∖ cdif 3898   ⊆ wss 3901  {csn 4580  {cpr 4582  ‘cfv 6492  Vtxcvtx 29069  Edgcedg 29120  USGraphcusgr 29222   FriendGraph cfrgr 30333

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 2708  ax-nul 5251

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-frgr 30334

This theorem is referenced by:  frgreu  30343  frcond3  30344  nfrgr2v  30347  3vfriswmgr  30353  2pthfrgrrn2  30358  2pthfrgr  30359  3cyclfrgrrn2  30362  3cyclfrgr  30363  n4cyclfrgr  30366  frgrnbnb  30368  vdgn0frgrv2  30370  vdgn1frgrv2  30371  frgrncvvdeqlem2  30375  frgrncvvdeqlem3  30376  frgrncvvdeqlem6  30379  frgrncvvdeqlem9  30382  frgrncvvdeq  30384  frgrwopreglem4a  30385  frgrwopreg  30398  frgrregorufrg  30401  frgr2wwlkeu  30402  frgr2wsp1  30405  frgr2wwlkeqm  30406  frrusgrord0lem  30414  frrusgrord0  30415  friendshipgt3  30473