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Theorem 1to3vfriswmgra 25735
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 986 . . 3  |-  ( ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } )  <->  ( ( V  =  { A }  \/  V  =  { A ,  B }
)  \/  V  =  { A ,  B ,  C } ) )
2 1to2vfriswmgra 25734 . . . . 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 437 . . . 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 4077 . . . . . . 7  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
54eqeq2d 2461 . . . . . 6  |-  ( B  =  C  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B } ) )
6 olc 386 . . . . . . . . . 10  |-  ( V  =  { A ,  B }  ->  ( V  =  { A }  \/  V  =  { A ,  B }
) )
76anim1i 572 . . . . . . . . 9  |-  ( ( V  =  { A ,  B }  /\  A  e.  X )  ->  (
( V  =  { A }  \/  V  =  { A ,  B } )  /\  A  e.  X ) )
87ancomd 453 . . . . . . . 8  |-  ( ( V  =  { A ,  B }  /\  A  e.  X )  ->  ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B } ) ) )
98, 2syl 17 . . . . . . 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 436 . . . . . 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 232 . . . . 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 4112 . . . . . . . 8  |-  ( -.  ( B  e.  _V  /\  B  =/=  A )  ->  { C ,  A ,  B }  =  { C ,  A } )
13 tprot 4067 . . . . . . . . . . . . 13  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
1413eqeq1i 2456 . . . . . . . . . . . 12  |-  ( { C ,  A ,  B }  =  { C ,  A }  <->  { A ,  B ,  C }  =  { C ,  A }
)
1514biimpi 198 . . . . . . . . . . 11  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { C ,  A }
)
16 prcom 4050 . . . . . . . . . . 11  |-  { C ,  A }  =  { A ,  C }
1715, 16syl6eq 2501 . . . . . . . . . 10  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { A ,  C }
)
1817eqeq2d 2461 . . . . . . . . 9  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  C }
) )
19 olc 386 . . . . . . . . . . 11  |-  ( V  =  { A ,  C }  ->  ( V  =  { A }  \/  V  =  { A ,  C }
) )
20 1to2vfriswmgra 25734 . . . . . . . . . . 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 477 . . . . . . . . . 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 437 . . . . . . . . 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 232 . . . . . . . 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 17 . . . . . . 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 26 . . . . . 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 4112 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  { B ,  A ,  C }  =  { B ,  A } )
27 tpcoma 4068 . . . . . . . . . . . . 13  |-  { B ,  A ,  C }  =  { A ,  B ,  C }
2827eqeq1i 2456 . . . . . . . . . . . 12  |-  ( { B ,  A ,  C }  =  { B ,  A }  <->  { A ,  B ,  C }  =  { B ,  A }
)
2928biimpi 198 . . . . . . . . . . 11  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { B ,  A }
)
30 prcom 4050 . . . . . . . . . . 11  |-  { B ,  A }  =  { A ,  B }
3129, 30syl6eq 2501 . . . . . . . . . 10  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { A ,  B }
)
3231eqeq2d 2461 . . . . . . . . 9  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B }
) )
336, 2sylan2 477 . . . . . . . . . . 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 437 . . . . . . . . . 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 26 . . . . . . . . 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 232 . . . . . . . 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 17 . . . . . . 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 81 . . . . . 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 459 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  e.  _V )
40 simpl 459 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  e.  _V )
4139, 40anim12i 570 . . . . . . . . . . . 12  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( B  e.  _V  /\  C  e.  _V ) )
4241ad2antrr 732 . . . . . . . . . . 11  |-  ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  /\  V  =  { A ,  B ,  C }
)  ->  ( B  e.  _V  /\  C  e. 
_V ) )
4342anim1i 572 . . . . . . . . . 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 453 . . . . . . . . 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 989 . . . . . . . . 9  |-  ( ( A  e.  X  /\  B  e.  _V  /\  C  e.  _V )  <->  ( A  e.  X  /\  ( B  e.  _V  /\  C  e.  _V ) ) )
4644, 45sylibr 216 . . . . . . . 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 463 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  =/=  A )
4847necomd 2679 . . . . . . . . . . . 12  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  A  =/=  B )
49 simpr 463 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  =/=  A )
5049necomd 2679 . . . . . . . . . . . 12  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  A  =/=  C )
5148, 50anim12i 570 . . . . . . . . . . 11  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( A  =/=  B  /\  A  =/=  C ) )
5251anim1i 572 . . . . . . . . . 10  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( ( A  =/= 
B  /\  A  =/=  C )  /\  B  =/= 
C ) )
53 df-3an 987 . . . . . . . . . 10  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( ( A  =/=  B  /\  A  =/=  C )  /\  B  =/=  C ) )
5452, 53sylibr 216 . . . . . . . . 9  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
5554ad2antrr 732 . . . . . . . 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 762 . . . . . . . 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 25733 . . . . . . . 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 1268 . . . . . . 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 615 . . . . . 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 953 . . . . 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 2707 . . . 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 381 . . 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 199 . 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 432 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 370    /\ wa 371    \/ w3o 984    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   E!wreu 2739   _Vcvv 3045    \ cdif 3401   {csn 3968   {cpr 3970   {ctp 3972   class class class wbr 4402   ran crn 4835   FriendGrph cfrgra 25716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516  df-usgra 25060  df-frgra 25717
This theorem is referenced by:  1to3vfriendship  25736
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