Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2pthfrgrarn Structured version   Unicode version

Theorem 2pthfrgrarn 28461
 Description: Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1 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.)
Assertion
Ref Expression
2pthfrgrarn FriendGrph
Distinct variable groups:   ,,,   ,,,

Proof of Theorem 2pthfrgrarn
StepHypRef Expression
1 frisusgrapr 28443 . 2 FriendGrph USGrph
2 reurex 2924 . . . . . . 7
3 prcom 3884 . . . . . . . . . . . 12
43eleq1i 2501 . . . . . . . . . . 11
54anbi1i 678 . . . . . . . . . 10
6 zfpair2 4406 . . . . . . . . . . 11
7 zfpair2 4406 . . . . . . . . . . 11
86, 7prss 3954 . . . . . . . . . 10
95, 8bitri 242 . . . . . . . . 9
109biimpri 199 . . . . . . . 8
1110reximi 2815 . . . . . . 7
122, 11syl 16 . . . . . 6
1312a1i 11 . . . . 5 USGrph
1413ralimdva 2786 . . . 4 USGrph
1514ralimdva 2786 . . 3 USGrph
1615imp 420 . 2 USGrph
171, 16syl 16 1 FriendGrph
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wcel 1726  wral 2707  wrex 2708  wreu 2709   cdif 3319   wss 3322  csn 3816  cpr 3817   class class class wbr 4214   crn 4881   USGrph cusg 21367   FriendGrph cfrgra 28440 This theorem is referenced by:  2pthfrgrarn2  28462  3cyclfrgrarn1  28464 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891  df-frgra 28441
 Copyright terms: Public domain W3C validator