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Theorem 1to3vfriswmgra 27762
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 937 . . 3  |-  ( ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } )  <->  ( ( V  =  { A }  \/  V  =  { A ,  B }
)  \/  V  =  { A ,  B ,  C } ) )
2 1to2vfriswmgra 27761 . . . . 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 425 . . . 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 3854 . . . . . . 7  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
54eqeq2d 2400 . . . . . 6  |-  ( B  =  C  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B } ) )
6 olc 374 . . . . . . . . . 10  |-  ( V  =  { A ,  B }  ->  ( V  =  { A }  \/  V  =  { A ,  B }
) )
76anim1i 552 . . . . . . . . 9  |-  ( ( V  =  { A ,  B }  /\  A  e.  X )  ->  (
( V  =  { A }  \/  V  =  { A ,  B } )  /\  A  e.  X ) )
87ancomd 439 . . . . . . . 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 424 . . . . . 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 220 . . . . 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 3883 . . . . . . . 8  |-  ( -.  ( B  e.  _V  /\  B  =/=  A )  ->  { C ,  A ,  B }  =  { C ,  A } )
13 tprot 3844 . . . . . . . . . . . . 13  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
1413eqeq1i 2396 . . . . . . . . . . . 12  |-  ( { C ,  A ,  B }  =  { C ,  A }  <->  { A ,  B ,  C }  =  { C ,  A }
)
1514biimpi 187 . . . . . . . . . . 11  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { C ,  A }
)
16 prcom 3827 . . . . . . . . . . 11  |-  { C ,  A }  =  { A ,  C }
1715, 16syl6eq 2437 . . . . . . . . . 10  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { A ,  C }
)
1817eqeq2d 2400 . . . . . . . . 9  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  C }
) )
19 olc 374 . . . . . . . . . . 11  |-  ( V  =  { A ,  C }  ->  ( V  =  { A }  \/  V  =  { A ,  C }
) )
20 1to2vfriswmgra 27761 . . . . . . . . . . 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 461 . . . . . . . . . 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 425 . . . . . . . . 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 220 . . . . . . . 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 23 . . . . . 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 3883 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  { B ,  A ,  C }  =  { B ,  A } )
27 tpcoma 3845 . . . . . . . . . . . . 13  |-  { B ,  A ,  C }  =  { A ,  B ,  C }
2827eqeq1i 2396 . . . . . . . . . . . 12  |-  ( { B ,  A ,  C }  =  { B ,  A }  <->  { A ,  B ,  C }  =  { B ,  A }
)
2928biimpi 187 . . . . . . . . . . 11  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { B ,  A }
)
30 prcom 3827 . . . . . . . . . . 11  |-  { B ,  A }  =  { A ,  B }
3129, 30syl6eq 2437 . . . . . . . . . 10  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { A ,  B }
)
3231eqeq2d 2400 . . . . . . . . 9  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B }
) )
336, 2sylan2 461 . . . . . . . . . . 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 425 . . . . . . . . . 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 23 . . . . . . . . 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 220 . . . . . . . 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 74 . . . . . 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 444 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  e.  _V )
40 simpl 444 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  e.  _V )
4139, 40anim12i 550 . . . . . . . . . . . 12  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( B  e.  _V  /\  C  e.  _V ) )
4241ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  /\  V  =  { A ,  B ,  C }
)  ->  ( B  e.  _V  /\  C  e. 
_V ) )
4342anim1i 552 . . . . . . . . . 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 439 . . . . . . . . 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 940 . . . . . . . . 9  |-  ( ( A  e.  X  /\  B  e.  _V  /\  C  e.  _V )  <->  ( A  e.  X  /\  ( B  e.  _V  /\  C  e.  _V ) ) )
4644, 45sylibr 204 . . . . . . . 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 448 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  =/=  A )
4847necomd 2635 . . . . . . . . . . . 12  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  A  =/=  B )
49 simpr 448 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  =/=  A )
5049necomd 2635 . . . . . . . . . . . 12  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  A  =/=  C )
5148, 50anim12i 550 . . . . . . . . . . 11  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( A  =/=  B  /\  A  =/=  C ) )
5251anim1i 552 . . . . . . . . . 10  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( ( A  =/= 
B  /\  A  =/=  C )  /\  B  =/= 
C ) )
53 df-3an 938 . . . . . . . . . 10  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( ( A  =/=  B  /\  A  =/=  C )  /\  B  =/=  C ) )
5452, 53sylibr 204 . . . . . . . . 9  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
5554ad2antrr 707 . . . . . . . 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 732 . . . . . . . 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 27760 . . . . . . . 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 1184 . . . . . . 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 594 . . . . . 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 909 . . . . 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 2628 . . . 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 369 . . 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 188 . 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 420 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 set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   E!wreu 2653   _Vcvv 2901    \ cdif 3262   {csn 3759   {cpr 3760   {ctp 3761   class class class wbr 4155   ran crn 4821   FriendGrph cfrgra 27743
This theorem is referenced by:  1to3vfriendship  27763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-hash 11548  df-usgra 21236  df-frgra 27744
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