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Theorem frgraeu 25627
Description: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Assertion
Ref Expression
frgraeu  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
Distinct variable groups:    A, b    C, b    E, b    V, b

Proof of Theorem frgraeu
StepHypRef Expression
1 frgraun 25569 . . . 4  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
21imp 430 . . 3  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C ) )  ->  E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )
3 df-reu 2789 . . . 4  |-  ( E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  <-> 
E! b ( b  e.  V  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
4 frisusgra 25565 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
5 usgraedgrnv 24950 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { A ,  b }  e.  ran  E )  ->  ( A  e.  V  /\  b  e.  V )
)
65simprd 464 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { A ,  b }  e.  ran  E )  ->  b  e.  V )
76expcom 436 . . . . . . . . . . 11  |-  ( { A ,  b }  e.  ran  E  -> 
( V USGrph  E  ->  b  e.  V ) )
87adantr 466 . . . . . . . . . 10  |-  ( ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  ->  ( V USGrph  E  ->  b  e.  V ) )
94, 8syl5com 31 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  -> 
b  e.  V ) )
109adantr 466 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C ) )  ->  ( ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  -> 
b  e.  V ) )
1110pm4.71rd 639 . . . . . . 7  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C ) )  ->  ( ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  <->  ( b  e.  V  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) ) )
1211bicomd 204 . . . . . 6  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C ) )  ->  ( ( b  e.  V  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )  <-> 
( { A , 
b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
1312eubidv 2288 . . . . 5  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C ) )  ->  ( E! b ( b  e.  V  /\  ( { A , 
b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )  <->  E! b
( { A , 
b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
1413biimpcd 227 . . . 4  |-  ( E! b ( b  e.  V  /\  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )  ->  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C ) )  ->  E! b ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
153, 14sylbi 198 . . 3  |-  ( E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  ->  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C ) )  ->  E! b ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
162, 15mpcom 37 . 2  |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C ) )  ->  E! b ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) )
1716ex 435 1  |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/= 
C )  ->  E! b ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1870   E!weu 2266    =/= wne 2625   E!wreu 2784   {cpr 4004   class class class wbr 4426   ran crn 4855   USGrph cusg 24903   FriendGrph cfrgra 25561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-hash 12513  df-usgra 24906  df-frgra 25562
This theorem is referenced by:  frg2woteqm  25632
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