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

Theorem 2pthfrgrarn 28433
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  |-  ( V FriendGrph  E  ->  A. a  e.  V  A. c  e.  ( V  \  { a } ) E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
Distinct variable groups:    E, a,
b, c    V, a,
b, c

Proof of Theorem 2pthfrgrarn
StepHypRef Expression
1 frisusgrapr 28418 . 2  |-  ( V FriendGrph  E  ->  ( V USGrph  E  /\  A. a  e.  V  A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_  ran  E ) )
2 reurex 2767 . . . . . . 7  |-  ( E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  E. b  e.  V  { { b ,  a } ,  { b ,  c } }  C_  ran  E )
3 prcom 3718 . . . . . . . . . . . 12  |-  { a ,  b }  =  { b ,  a }
43eleq1i 2359 . . . . . . . . . . 11  |-  ( { a ,  b }  e.  ran  E  <->  { b ,  a }  e.  ran  E )
54anbi1i 676 . . . . . . . . . 10  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
6 zfpair2 4231 . . . . . . . . . . 11  |-  { b ,  a }  e.  _V
7 zfpair2 4231 . . . . . . . . . . 11  |-  { b ,  c }  e.  _V
86, 7prss 3785 . . . . . . . . . 10  |-  ( ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
95, 8bitri 240 . . . . . . . . 9  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
109biimpri 197 . . . . . . . 8  |-  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
1110reximi 2663 . . . . . . 7  |-  ( E. b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  E. b  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
122, 11syl 15 . . . . . 6  |-  ( E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  E. b  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
1312a1i 10 . . . . 5  |-  ( ( ( V USGrph  E  /\  a  e.  V )  /\  c  e.  ( V  \  { a } ) )  ->  ( E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  E. b  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) ) )
1413ralimdva 2634 . . . 4  |-  ( ( V USGrph  E  /\  a  e.  V )  ->  ( A. c  e.  ( V  \  { a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_  ran  E  ->  A. c  e.  ( V  \  { a } ) E. b  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) ) )
1514ralimdva 2634 . . 3  |-  ( V USGrph  E  ->  ( A. a  e.  V  A. c  e.  ( V  \  {
a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  A. a  e.  V  A. c  e.  ( V  \  {
a } ) E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
) ) )
1615imp 418 . 2  |-  ( ( V USGrph  E  /\  A. a  e.  V  A. c  e.  ( V  \  {
a } ) E! b  e.  V  { { b ,  a } ,  { b ,  c } }  C_ 
ran  E )  ->  A. a  e.  V  A. c  e.  ( V  \  { a } ) E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
171, 16syl 15 1  |-  ( V FriendGrph  E  ->  A. a  e.  V  A. c  e.  ( V  \  { a } ) E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558    \ cdif 3162    C_ wss 3165   {csn 3653   {cpr 3654   class class class wbr 4039   ran crn 4706   USGrph cusg 28227   FriendGrph cfrgra 28415
This theorem is referenced by:  2pthfrgrarn2  28434  3cyclfrgrarn1  28435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-frgra 28416
  Copyright terms: Public domain W3C validator