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Theorem 2pthfrgrarn 25786
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.)
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 25768 . 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 3021 . . . . . . 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 4063 . . . . . . . . . . . 12  |-  { a ,  b }  =  { b ,  a }
43eleq1i 2531 . . . . . . . . . . 11  |-  ( { a ,  b }  e.  ran  E  <->  { b ,  a }  e.  ran  E )
54anbi1i 706 . . . . . . . . . 10  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
6 zfpair2 4654 . . . . . . . . . . 11  |-  { b ,  a }  e.  _V
7 zfpair2 4654 . . . . . . . . . . 11  |-  { b ,  c }  e.  _V
86, 7prss 4139 . . . . . . . . . 10  |-  ( ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
95, 8bitri 257 . . . . . . . . 9  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
109biimpri 211 . . . . . . . 8  |-  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
1110reximi 2867 . . . . . . 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 17 . . . . . 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 2808 . . . 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 2808 . . 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 435 . 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 17 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 setvar class
Syntax hints:    -> wi 4    /\ wa 375    e. wcel 1898   A.wral 2749   E.wrex 2750   E!wreu 2751    \ cdif 3413    C_ wss 3416   {csn 3980   {cpr 3982   class class class wbr 4416   ran crn 4854   USGrph cusg 25106   FriendGrph cfrgra 25765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-xp 4859  df-rel 4860  df-cnv 4861  df-dm 4863  df-rn 4864  df-frgra 25766
This theorem is referenced by:  2pthfrgrarn2  25787  3cyclfrgrarn1  25789
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