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Theorem frg2woteu 30791
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 988 . . 3  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  V FriendGrph  E )
2 simpl 457 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V )  ->  A  e.  V )
32adantr 465 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  A  e.  V )
4 simpr 461 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V )  ->  B  e.  V )
54adantr 465 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  B  e.  V )
6 simpr 461 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  A  =/=  B )
73, 5, 63jca 1168 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  A  =/=  B )  ->  ( A  e.  V  /\  B  e.  V  /\  A  =/= 
B ) )
873adant1 1006 . . 3  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B
)  ->  ( A  e.  V  /\  B  e.  V  /\  A  =/= 
B ) )
9 frgraun 30731 . . 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 60 . 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 30727 . . . . . . 7  |-  ( V FriendGrph  E  ->  V USGrph  E )
1211adantr 465 . . . . . 6  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V )
)  ->  V USGrph  E )
1312adantr 465 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  V USGrph  E )
142ad2antlr 726 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  A  e.  V )
15 simpr 461 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  c  e.  V )
164ad2antlr 726 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V
) )  /\  c  e.  V )  ->  B  e.  V )
17 usg2wlkonot 30545 . . . . 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 1221 . . . 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 3004 . . 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 1008 . 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 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2645   E!wreu 2798   {cpr 3982   <.cotp 3988   class class class wbr 4395   ran crn 4944  (class class class)co 6195   USGrph cusg 23411   2WalksOnOt c2wlkonot 30517   FriendGrph cfrgra 30723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-ot 3989  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-fzo 11661  df-hash 12216  df-word 12342  df-usgra 23413  df-wlk 23562  df-wlkon 23568  df-2wlkonot 30520  df-frgra 30724
This theorem is referenced by:  frg2wotn0  30792  frg2wot1  30793
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