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Theorem frg2woteu 28444
Description: For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices as ordered triple. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Assertion
Ref Expression
frg2woteu  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  E! c  e.  V  <. A , 
c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B ) )
Distinct variable groups:    A, c    B, c    E, c    V, c

Proof of Theorem frg2woteu
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  V FriendGrph  E )
2 simpl 444 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
32adantr 452 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  A  e.  V )
4 simpr 448 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V )  ->  B  e.  V )
54adantr 452 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  B  e.  V )
6 simpr 448 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  A  =/=  B )
73, 5, 63jca 1134 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( A  e.  V  /\  B  e.  V  /\  A  =/= 
B ) )
873adant1 975 . . 3  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A  e.  V  /\  B  e.  V  /\  A  =/= 
B ) )
9 frgraun 28386 . . 3  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  B  e.  V  /\  A  =/= 
B )  ->  E! c  e.  V  ( { A ,  c }  e.  ran  E  /\  { c ,  B }  e.  ran  E ) ) )
101, 8, 9sylc 58 . 2  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  E! c  e.  V  ( { A ,  c }  e.  ran  E  /\  {
c ,  B }  e.  ran  E ) )
11 frisusgra 28382 . . . . . . 7  |-  ( V FriendGrph  E  ->  V USGrph  E )
1211adantr 452 . . . . . 6  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )
)  ->  V USGrph  E )
1312adantr 452 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  V USGrph  E )
142ad2antlr 708 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  A  e.  V )
15 simpr 448 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  c  e.  V )
164ad2antlr 708 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  B  e.  V )
17 usg2wlkonot 28350 . . . . 5  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  c  e.  V  /\  B  e.  V )
)  ->  ( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  <-> 
( { A , 
c }  e.  ran  E  /\  { c ,  B }  e.  ran  E ) ) )
1813, 14, 15, 16, 17syl13anc 1186 . . . 4  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  ( <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  <->  ( { A ,  c }  e.  ran  E  /\  {
c ,  B }  e.  ran  E ) ) )
1918reubidva 2891 . . 3  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )
)  ->  ( E! c  e.  V  <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  <-> 
E! c  e.  V  ( { A ,  c }  e.  ran  E  /\  { c ,  B }  e.  ran  E ) ) )
20193adant3 977 . 2  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( E! c  e.  V  <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B )  <-> 
E! c  e.  V  ( { A ,  c }  e.  ran  E  /\  { c ,  B }  e.  ran  E ) ) )
2110, 20mpbird 224 1  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  E! c  e.  V  <. A , 
c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2599   E!wreu 2707   {cpr 3815   <.cotp 3818   class class class wbr 4212   ran crn 4879  (class class class)co 6081   USGrph cusg 21365   2WalksOnOt c2wlkonot 28322   FriendGrph cfrgra 28378
This theorem is referenced by:  frg2wotn0  28445  frg2wot1  28446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-usgra 21367  df-wlk 21516  df-wlkon 21522  df-2wlkonot 28325  df-frgra 28379
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