Theorem frgrusgrfrcond 27123

Description: A friendship graph is a simple graph which fulfils the friendship condition. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
isfrgr.v	|- V = (Vtx ` G )
isfrgr.e	|- E = (Edg ` G )

Assertion

Ref	Expression
frgrusgrfrcond	|- ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) )

Distinct variable groups:   k, l, x, G   k, V, l, x

Allowed substitution hints:   E( x, k, l)

Proof of Theorem frgrusgrfrcond

Step	Hyp	Ref	Expression
1		isfrgr.v	. . . . 5 |- V = (Vtx ` G )
2		isfrgr.e	. . . . 5 |- E = (Edg ` G )
3	1, 2	isfrgr 27122	. . . 4 |- ( G e. FriendGraph -> ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) ) )
4		simpl 473	. . . 4 |- ( ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) -> G e. USGraph )
5	3, 4	syl6bi 243	. . 3 |- ( G e. FriendGraph -> ( G e. FriendGraph -> G e. USGraph ) )
6	5	pm2.43i 52	. 2 |- ( G e. FriendGraph -> G e. USGraph )
7	1, 2	isfrgr 27122	. 2 |- ( G e. USGraph -> ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) ) )
8	6, 4, 7	pm5.21nii 368	1 |- ( G e. FriendGraph <-> ( G e. USGraph /\ A. k e. V A. l e. ( V \ { k } ) E! x e. V { { x , k } , { x , l } } C_ E ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   \ cdif 3571   C_ 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:  frgrusgr  27124  frgr0v  27125  frgr0  27128  frcond1  27130  frgr1v  27135  nfrgr2v  27136  frgr3v  27139  2pthfrgrrn  27146  n4cyclfrgr  27155