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Theorem 1to3vfriswmgra 24985
Description: Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
1to3vfriswmgra  |-  ( ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } ) )  -> 
( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
Distinct variable groups:    A, h, v, w    B, h, v, w    C, h, v, w   
h, E, v, w   
h, V, v, w   
v, X, w
Allowed substitution hint:    X( h)

Proof of Theorem 1to3vfriswmgra
StepHypRef Expression
1 df-3or 975 . . 3  |-  ( ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } )  <->  ( ( V  =  { A }  \/  V  =  { A ,  B }
)  \/  V  =  { A ,  B ,  C } ) )
2 1to2vfriswmgra 24984 . . . . 5  |-  ( ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B } ) )  -> 
( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
32expcom 435 . . . 4  |-  ( ( V  =  { A }  \/  V  =  { A ,  B }
)  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h }
) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E ) ) ) )
4 tppreq3 4120 . . . . . . 7  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
54eqeq2d 2457 . . . . . 6  |-  ( B  =  C  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B } ) )
6 olc 384 . . . . . . . . . 10  |-  ( V  =  { A ,  B }  ->  ( V  =  { A }  \/  V  =  { A ,  B }
) )
76anim1i 568 . . . . . . . . 9  |-  ( ( V  =  { A ,  B }  /\  A  e.  X )  ->  (
( V  =  { A }  \/  V  =  { A ,  B } )  /\  A  e.  X ) )
87ancomd 451 . . . . . . . 8  |-  ( ( V  =  { A ,  B }  /\  A  e.  X )  ->  ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B } ) ) )
98, 2syl 16 . . . . . . 7  |-  ( ( V  =  { A ,  B }  /\  A  e.  X )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
109ex 434 . . . . . 6  |-  ( V  =  { A ,  B }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) )
115, 10syl6bi 228 . . . . 5  |-  ( B  =  C  ->  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E ) ) ) ) )
12 tpprceq3 4155 . . . . . . . 8  |-  ( -.  ( B  e.  _V  /\  B  =/=  A )  ->  { C ,  A ,  B }  =  { C ,  A } )
13 tprot 4110 . . . . . . . . . . . . 13  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
1413eqeq1i 2450 . . . . . . . . . . . 12  |-  ( { C ,  A ,  B }  =  { C ,  A }  <->  { A ,  B ,  C }  =  { C ,  A }
)
1514biimpi 194 . . . . . . . . . . 11  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { C ,  A }
)
16 prcom 4093 . . . . . . . . . . 11  |-  { C ,  A }  =  { A ,  C }
1715, 16syl6eq 2500 . . . . . . . . . 10  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { A ,  C }
)
1817eqeq2d 2457 . . . . . . . . 9  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  C }
) )
19 olc 384 . . . . . . . . . . 11  |-  ( V  =  { A ,  C }  ->  ( V  =  { A }  \/  V  =  { A ,  C }
) )
20 1to2vfriswmgra 24984 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  C } ) )  -> 
( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
2119, 20sylan2 474 . . . . . . . . . 10  |-  ( ( A  e.  X  /\  V  =  { A ,  C } )  -> 
( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
2221expcom 435 . . . . . . . . 9  |-  ( V  =  { A ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) )
2318, 22syl6bi 228 . . . . . . . 8  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) ) )
2412, 23syl 16 . . . . . . 7  |-  ( -.  ( B  e.  _V  /\  B  =/=  A )  ->  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) ) )
2524a1d 25 . . . . . 6  |-  ( -.  ( B  e.  _V  /\  B  =/=  A )  ->  ( B  =/= 
C  ->  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) ) ) )
26 tpprceq3 4155 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  { B ,  A ,  C }  =  { B ,  A } )
27 tpcoma 4111 . . . . . . . . . . . . 13  |-  { B ,  A ,  C }  =  { A ,  B ,  C }
2827eqeq1i 2450 . . . . . . . . . . . 12  |-  ( { B ,  A ,  C }  =  { B ,  A }  <->  { A ,  B ,  C }  =  { B ,  A }
)
2928biimpi 194 . . . . . . . . . . 11  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { B ,  A }
)
30 prcom 4093 . . . . . . . . . . 11  |-  { B ,  A }  =  { A ,  B }
3129, 30syl6eq 2500 . . . . . . . . . 10  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { A ,  B }
)
3231eqeq2d 2457 . . . . . . . . 9  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B }
) )
336, 2sylan2 474 . . . . . . . . . . 11  |-  ( ( A  e.  X  /\  V  =  { A ,  B } )  -> 
( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
3433expcom 435 . . . . . . . . . 10  |-  ( V  =  { A ,  B }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) )
3534a1d 25 . . . . . . . . 9  |-  ( V  =  { A ,  B }  ->  ( B  =/=  C  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) ) )
3632, 35syl6bi 228 . . . . . . . 8  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  ( V  =  { A ,  B ,  C }  ->  ( B  =/=  C  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) ) ) )
3726, 36syl 16 . . . . . . 7  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  ( V  =  { A ,  B ,  C }  ->  ( B  =/=  C  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) ) ) )
3837com23 78 . . . . . 6  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  ( B  =/= 
C  ->  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) ) ) )
39 simpl 457 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  e.  _V )
40 simpl 457 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  e.  _V )
4139, 40anim12i 566 . . . . . . . . . . . 12  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( B  e.  _V  /\  C  e.  _V ) )
4241ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  /\  V  =  { A ,  B ,  C }
)  ->  ( B  e.  _V  /\  C  e. 
_V ) )
4342anim1i 568 . . . . . . . . . 10  |-  ( ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/= 
C )  /\  V  =  { A ,  B ,  C } )  /\  A  e.  X )  ->  ( ( B  e. 
_V  /\  C  e.  _V )  /\  A  e.  X ) )
4443ancomd 451 . . . . . . . . 9  |-  ( ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/= 
C )  /\  V  =  { A ,  B ,  C } )  /\  A  e.  X )  ->  ( A  e.  X  /\  ( B  e.  _V  /\  C  e.  _V )
) )
45 3anass 978 . . . . . . . . 9  |-  ( ( A  e.  X  /\  B  e.  _V  /\  C  e.  _V )  <->  ( A  e.  X  /\  ( B  e.  _V  /\  C  e.  _V ) ) )
4644, 45sylibr 212 . . . . . . . 8  |-  ( ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/= 
C )  /\  V  =  { A ,  B ,  C } )  /\  A  e.  X )  ->  ( A  e.  X  /\  B  e.  _V  /\  C  e.  _V )
)
47 simpr 461 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  =/=  A )
4847necomd 2714 . . . . . . . . . . . 12  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  A  =/=  B )
49 simpr 461 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  =/=  A )
5049necomd 2714 . . . . . . . . . . . 12  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  A  =/=  C )
5148, 50anim12i 566 . . . . . . . . . . 11  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( A  =/=  B  /\  A  =/=  C ) )
5251anim1i 568 . . . . . . . . . 10  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( ( A  =/= 
B  /\  A  =/=  C )  /\  B  =/= 
C ) )
53 df-3an 976 . . . . . . . . . 10  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( ( A  =/=  B  /\  A  =/=  C )  /\  B  =/=  C ) )
5452, 53sylibr 212 . . . . . . . . 9  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
5554ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/= 
C )  /\  V  =  { A ,  B ,  C } )  /\  A  e.  X )  ->  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
56 simplr 755 . . . . . . . 8  |-  ( ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/= 
C )  /\  V  =  { A ,  B ,  C } )  /\  A  e.  X )  ->  V  =  { A ,  B ,  C }
)
57 3vfriswmgra 24983 . . . . . . . 8  |-  ( ( ( A  e.  X  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
)  /\  V  =  { A ,  B ,  C } )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
5846, 55, 56, 57syl3anc 1229 . . . . . . 7  |-  ( ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/= 
C )  /\  V  =  { A ,  B ,  C } )  /\  A  e.  X )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E ) ) )
5958exp41 610 . . . . . 6  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( B  =/=  C  ->  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E ) ) ) ) ) )
6025, 38, 59ecase 942 . . . . 5  |-  ( B  =/=  C  ->  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E ) ) ) ) )
6111, 60pm2.61ine 2756 . . . 4  |-  ( V  =  { A ,  B ,  C }  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E ) ) ) )
623, 61jaoi 379 . . 3  |-  ( ( ( V  =  { A }  \/  V  =  { A ,  B } )  \/  V  =  { A ,  B ,  C } )  -> 
( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E ) ) ) )
631, 62sylbi 195 . 2  |-  ( ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } )  ->  ( A  e.  X  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) ) )
6463impcom 430 1  |-  ( ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } ) )  -> 
( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 973    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   E!wreu 2795   _Vcvv 3095    \ cdif 3458   {csn 4014   {cpr 4016   {ctp 4018   class class class wbr 4437   ran crn 4990   FriendGrph cfrgra 24966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388  df-usgra 24311  df-frgra 24967
This theorem is referenced by:  1to3vfriendship  24986
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