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Theorem frgraeu 24717
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 24658 . . . 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 429 . . 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 2814 . . . 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 24654 . . . . . . . . . 10  |-  ( V FriendGrph  E  ->  V USGrph  E )
5 usgraedgrnv 24039 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { A ,  b }  e.  ran  E )  ->  ( A  e.  V  /\  b  e.  V )
)
65simprd 463 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  { A ,  b }  e.  ran  E )  ->  b  e.  V )
76expcom 435 . . . . . . . . . . 11  |-  ( { A ,  b }  e.  ran  E  -> 
( V USGrph  E  ->  b  e.  V ) )
87adantr 465 . . . . . . . . . 10  |-  ( ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  ->  ( V USGrph  E  ->  b  e.  V ) )
94, 8syl5com 30 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E )  -> 
b  e.  V ) )
109adantr 465 . . . . . . . 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 635 . . . . . . 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 201 . . . . . 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 2291 . . . . 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 224 . . . 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 195 . . 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 36 . 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 434 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 369    /\ w3a 968    e. wcel 1762   E!weu 2268    =/= wne 2655   E!wreu 2809   {cpr 4022   class class class wbr 4440   ran crn 4993   USGrph cusg 23993   FriendGrph cfrgra 24650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-hash 12361  df-usgra 23996  df-frgra 24651
This theorem is referenced by:  frg2woteqm  24722
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