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Theorem 2pthfrgrarn 25136
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 25118 . 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 3074 . . . . . . 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 4110 . . . . . . . . . . . 12  |-  { a ,  b }  =  { b ,  a }
43eleq1i 2534 . . . . . . . . . . 11  |-  ( { a ,  b }  e.  ran  E  <->  { b ,  a }  e.  ran  E )
54anbi1i 695 . . . . . . . . . 10  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
6 zfpair2 4696 . . . . . . . . . . 11  |-  { b ,  a }  e.  _V
7 zfpair2 4696 . . . . . . . . . . 11  |-  { b ,  c }  e.  _V
86, 7prss 4186 . . . . . . . . . 10  |-  ( ( { b ,  a }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
95, 8bitri 249 . . . . . . . . 9  |-  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  <->  { { b ,  a } ,  {
b ,  c } }  C_  ran  E )
109biimpri 206 . . . . . . . 8  |-  ( { { b ,  a } ,  { b ,  c } }  C_ 
ran  E  ->  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
1110reximi 2925 . . . . . . 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 2865 . . . 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 2865 . . 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 429 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809    \ cdif 3468    C_ wss 3471   {csn 4032   {cpr 4034   class class class wbr 4456   ran crn 5009   USGrph cusg 24457   FriendGrph cfrgra 25115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019  df-frgra 25116
This theorem is referenced by:  2pthfrgrarn2  25137  3cyclfrgrarn1  25139
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