Theorem frgrusgr 28198

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 2738	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2738	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	isfrgr 28197	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 501	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3053  ∃!wreu 3055   ∖ cdif 3840   ⊆ wss 3843  {csn 4516  {cpr 4518  ‘cfv 6339  Vtxcvtx 26941  Edgcedg 26992  USGraphcusgr 27094   FriendGraph cfrgr 28195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-iota 6297  df-fv 6347  df-frgr 28196

This theorem is referenced by:  frgreu  28205  frcond3  28206  nfrgr2v  28209  3vfriswmgr  28215  2pthfrgrrn2  28220  2pthfrgr  28221  3cyclfrgrrn2  28224  3cyclfrgr  28225  n4cyclfrgr  28228  frgrnbnb  28230  vdgn0frgrv2  28232  vdgn1frgrv2  28233  frgrncvvdeqlem2  28237  frgrncvvdeqlem3  28238  frgrncvvdeqlem6  28241  frgrncvvdeqlem9  28244  frgrncvvdeq  28246  frgrwopreglem4a  28247  frgrwopreg  28260  frgrregorufrg  28263  frgr2wwlkeu  28264  frgr2wsp1  28267  frgr2wwlkeqm  28268  frrusgrord0lem  28276  frrusgrord0  28277  friendshipgt3  28335