Theorem 2pthfrgrrn 27146

Description: Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.) (Revised by AV, 1-Apr-2021.)

Hypotheses

Ref	Expression
2pthfrgrrn.v	|- V = (Vtx ` G )
2pthfrgrrn.e	|- E = (Edg ` G )

Assertion

Ref	Expression
2pthfrgrrn	|- ( G e. FriendGraph -> A. a e. V A. c e. ( V \ { a } ) E. b e. V ( { a , b } e. E /\ { b , c } e. E ) )

Distinct variable groups:   G, a, b, c   V, a, b, c

Allowed substitution hints:   E( a, b, c)

Proof of Theorem 2pthfrgrrn

Step	Hyp	Ref	Expression
1		2pthfrgrrn.v	. . 3 |- V = (Vtx ` G )
2		2pthfrgrrn.e	. . 3 |- E = (Edg ` G )
3	1, 2	frgrusgrfrcond 27123	. 2 |- ( G e. FriendGraph <-> ( G e. USGraph /\ A. a e. V A. c e. ( V \ { a } ) E! b e. V { { b , a } , { b , c } } C_ E ) )
4		reurex 3160	. . . . . 6 |- ( E! b e. V { { b , a } , { b , c } } C_ E -> E. b e. V { { b , a } , { b , c } } C_ E )
5		prcom 4267	. . . . . . . . . 10 |- { a , b } = { b , a }
6	5	eleq1i 2692	. . . . . . . . 9 |- ( { a , b } e. E <-> { b , a } e. E )
7	6	anbi1i 731	. . . . . . . 8 |- ( ( { a , b } e. E /\ { b , c } e. E ) <-> ( { b , a } e. E /\ { b , c } e. E ) )
8		zfpair2 4907	. . . . . . . . 9 |- { b , a } e. _V
9		zfpair2 4907	. . . . . . . . 9 |- { b , c } e. _V
10	8, 9	prss 4351	. . . . . . . 8 |- ( ( { b , a } e. E /\ { b , c } e. E ) <-> { { b , a } , { b , c } } C_ E )
11	7, 10	sylbbr 226	. . . . . . 7 |- ( { { b , a } , { b , c } } C_ E -> ( { a , b } e. E /\ { b , c } e. E ) )
12	11	reximi 3011	. . . . . 6 |- ( E. b e. V { { b , a } , { b , c } } C_ E -> E. b e. V ( { a , b } e. E /\ { b , c } e. E ) )
13	4, 12	syl 17	. . . . 5 |- ( E! b e. V { { b , a } , { b , c } } C_ E -> E. b e. V ( { a , b } e. E /\ { b , c } e. E ) )
14	13	a1i 11	. . . 4 |- ( ( G e. USGraph /\ ( a e. V /\ c e. ( V \ { a } ) ) ) -> ( E! b e. V { { b , a } , { b , c } } C_ E -> E. b e. V ( { a , b } e. E /\ { b , c } e. E ) ) )
15	14	ralimdvva 2964	. . 3 |- ( G e. USGraph -> ( A. a e. V A. c e. ( V \ { a } ) E! b e. V { { b , a } , { b , c } } C_ E -> A. a e. V A. c e. ( V \ { a } ) E. b e. V ( { a , b } e. E /\ { b , c } e. E ) ) )
16	15	imp 445	. 2 |- ( ( G e. USGraph /\ A. a e. V A. c e. ( V \ { a } ) E! b e. V { { b , a } , { b , c } } C_ E ) -> A. a e. V A. c e. ( V \ { a } ) E. b e. V ( { a , b } e. E /\ { b , c } e. E ) )
17	3, 16	sylbi 207	1 |- ( G e. FriendGraph -> A. a e. V A. c e. ( V \ { a } ) E. b e. V ( { a , b } e. E /\ { b , c } e. E ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   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-sep 4781  ax-nul 4789  ax-pr 4906

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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:  2pthfrgrrn2  27147  3cyclfrgrrn1  27149