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Theorem 1to3vfriswmgra 30599
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 966 . . 3  |-  ( ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } )  <->  ( ( V  =  { A }  \/  V  =  { A ,  B }
)  \/  V  =  { A ,  B ,  C } ) )
2 1to2vfriswmgra 30598 . . . . 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 3980 . . . . . . 7  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
54eqeq2d 2454 . . . . . 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 4013 . . . . . . . 8  |-  ( -.  ( B  e.  _V  /\  B  =/=  A )  ->  { C ,  A ,  B }  =  { C ,  A } )
13 tprot 3970 . . . . . . . . . . . . 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 3953 . . . . . . . . . . 11  |-  { C ,  A }  =  { A ,  C }
1715, 16syl6eq 2491 . . . . . . . . . 10  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { A ,  C }
)
1817eqeq2d 2454 . . . . . . . . 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 30598 . . . . . . . . . . 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 4013 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  { B ,  A ,  C }  =  { B ,  A } )
27 tpcoma 3971 . . . . . . . . . . . . 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 3953 . . . . . . . . . . 11  |-  { B ,  A }  =  { A ,  B }
3129, 30syl6eq 2491 . . . . . . . . . 10  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { A ,  B }
)
3231eqeq2d 2454 . . . . . . . . 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 969 . . . . . . . . 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 2695 . . . . . . . . . . . 12  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  A  =/=  B )
49 simpr 461 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  =/=  A )
5049necomd 2695 . . . . . . . . . . . 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 967 . . . . . . . . . 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 754 . . . . . . . 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 30597 . . . . . . . 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 1218 . . . . . . 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 933 . . . . 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 2687 . . . 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 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   E!wreu 2717   _Vcvv 2972    \ cdif 3325   {csn 3877   {cpr 3879   {ctp 3881   class class class wbr 4292   ran crn 4841   FriendGrph cfrgra 30580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-hash 12104  df-usgra 23266  df-frgra 30581
This theorem is referenced by:  1to3vfriendship  30600
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