Theorem frgrusgr 27124

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 2622	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2622	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	frgrusgrfrcond 27123	. 2 ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
4	3	simplbi 476	1 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )

Syntax hints:   → wi 4   ∈ wcel 1990  ∀wral 2912  ∃!wreu 2914   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  {cpr 4179  ‘cfv 5888  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-frgr 27121

This theorem is referenced by:  frgreu  27132  frcond3  27133  nfrgr2v  27136  3vfriswmgr  27142  2pthfrgrrn2  27147  2pthfrgr  27148  3cyclfrgrrn2  27151  3cyclfrgr  27152  n4cyclfrgr  27155  frgrnbnb  27157  vdgn0frgrv2  27159  vdgn1frgrv2  27160  frgrncvvdeqlem2  27164  frgrncvvdeqlem3  27165  frgrncvvdeqlem6  27168  frgrncvvdeqlem9  27171  frgrncvvdeq  27173  frgrwopreglem4a  27174  frgrwopreg  27187  frgrregorufrg  27190  frgr2wwlkeu  27191  frgr2wsp1  27194  frgr2wwlkeqm  27195  frrusgrord0lem  27203  frrusgrord0  27204  friendshipgt3  27256