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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  |-  ( 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 28443 . 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 2924 . . . . . . 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 3884 . . . . . . . . . . . 12  |-  { a ,  b }  =  { b ,  a }
43eleq1i 2501 . . . . . . . . . . 11  |-  ( { a ,  b }  e.  ran  E  <->  { b ,  a }  e.  ran  E )
54anbi1i 678 . . . . . . . . . 10  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
6 zfpair2 4406 . . . . . . . . . . 11  |-  { b ,  a }  e.  _V
7 zfpair2 4406 . . . . . . . . . . 11  |-  { b ,  c }  e.  _V
86, 7prss 3954 . . . . . . . . . 10  |-  ( ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
95, 8bitri 242 . . . . . . . . 9  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
109biimpri 199 . . . . . . . 8  |-  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
1110reximi 2815 . . . . . . 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 16 . . . . . 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 11 . . . . 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 2786 . . . 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 2786 . . 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 420 . 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 16 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 360    e. wcel 1726   A.wral 2707   E.wrex 2708   E!wreu 2709    \ cdif 3319    C_ wss 3322   {csn 3816   {cpr 3817   class class class wbr 4214   ran 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
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