Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2pthfrgrarn2 Structured version   Unicode version

Theorem 2pthfrgrarn2 28402
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, 16-Nov-2017.)
Assertion
Ref Expression
2pthfrgrarn2  |-  ( 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 )  /\  ( a  =/=  b  /\  b  =/=  c ) ) )
Distinct variable groups:    E, a,
b, c    V, a,
b, c

Proof of Theorem 2pthfrgrarn2
StepHypRef Expression
1 2pthfrgrarn 28401 . 2  |-  ( 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 ) )
2 frisusgra 28384 . . . . . . . 8  |-  ( V FriendGrph  E  ->  V USGrph  E )
3 usgraedgrn 21403 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  { a ,  b }  e.  ran  E )  ->  a  =/=  b )
43ex 425 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( { a ,  b }  e.  ran  E  ->  a  =/=  b ) )
5 usgraedgrn 21403 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  { b ,  c }  e.  ran  E )  ->  b  =/=  c )
65ex 425 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( { b ,  c }  e.  ran  E  ->  b  =/=  c ) )
74, 6anim12d 548 . . . . . . . 8  |-  ( V USGrph  E  ->  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  ->  ( a  =/=  b  /\  b  =/=  c ) ) )
82, 7syl 16 . . . . . . 7  |-  ( V FriendGrph  E  ->  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  ->  ( a  =/=  b  /\  b  =/=  c ) ) )
98ad3antrrr 712 . . . . . 6  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  c  e.  ( V  \  { a } ) )  /\  b  e.  V )  ->  (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  ->  ( a  =/=  b  /\  b  =/=  c ) ) )
109ancld 538 . . . . 5  |-  ( ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  c  e.  ( V  \  { a } ) )  /\  b  e.  V )  ->  (
( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  ->  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  /\  ( a  =/=  b  /\  b  =/=  c ) ) ) )
1110reximdva 2820 . . . 4  |-  ( ( ( V FriendGrph  E  /\  a  e.  V )  /\  c  e.  ( V  \  { a } ) )  ->  ( E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  ->  E. b  e.  V  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E
)  /\  ( a  =/=  b  /\  b  =/=  c ) ) ) )
1211ralimdva 2786 . . 3  |-  ( ( V FriendGrph  E  /\  a  e.  V )  ->  ( A. c  e.  ( V  \  { a } ) E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  ->  A. c  e.  ( V  \  {
a } ) E. b  e.  V  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( a  =/=  b  /\  b  =/=  c ) ) ) )
1312ralimdva 2786 . 2  |-  ( 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
)  ->  A. a  e.  V  A. c  e.  ( V  \  {
a } ) E. b  e.  V  ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( a  =/=  b  /\  b  =/=  c ) ) ) )
141, 13mpd 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 )  /\  ( a  =/=  b  /\  b  =/=  c ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    \ cdif 3319   {csn 3816   {cpr 3817   class class class wbr 4214   ran crn 4881   USGrph cusg 21367   FriendGrph cfrgra 28380
This theorem is referenced by:  2pthfrgra  28403  3cyclfrgrarn1  28404
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-hash 11621  df-usgra 21369  df-frgra 28381
  Copyright terms: Public domain W3C validator