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Theorem frgrusgr 27003
 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.
StepHypRef Expression
1 eqid 2621 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2621 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2frgrusgrfrcond 27002 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
43simplbi 476 1 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  ∀wral 2907  ∃!wreu 2909   ∖ cdif 3556   ⊆ wss 3559  {csn 4153  {cpr 4155  ‘cfv 5852  Vtxcvtx 25787  Edgcedg 25852   USGraph cusgr 25950   FriendGraph cfrgr 26999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-frgr 27000 This theorem is referenced by:  nfrgr2v  27013  3vfriswmgr  27019  2pthfrgrrn2  27024  2pthfrgr  27025  3cyclfrgrrn2  27028  3cyclfrgr  27029  n4cyclfrgr  27032  frgrnbnb  27034  vdgn0frgrv2  27036  vdgn1frgrv2  27037  frgrncvvdeqlem2  27041  frgrncvvdeqlem3  27042  frgrncvvdeqlem4  27043  frgrncvvdeqlem7  27046  frgrncvvdeqlemC  27049  frgrncvvdeq  27051  frgrwopreglem4  27055  frgrwopreg  27057  frgreu  27062  frgr2wwlkeu  27063  frgr2wsp1  27066  frgr2wwlkeqm  27067  frrusgrord0  27074  frgrregorufrg  27076  friendshipgt3  27123
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