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Theorem 1to3vfriswmgra 25721
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 983 . . 3  |-  ( ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } )  <->  ( ( V  =  { A }  \/  V  =  { A ,  B }
)  \/  V  =  { A ,  B ,  C } ) )
2 1to2vfriswmgra 25720 . . . . 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 436 . . . 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 4102 . . . . . . 7  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
54eqeq2d 2436 . . . . . 6  |-  ( B  =  C  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B } ) )
6 olc 385 . . . . . . . . . 10  |-  ( V  =  { A ,  B }  ->  ( V  =  { A }  \/  V  =  { A ,  B }
) )
76anim1i 570 . . . . . . . . 9  |-  ( ( V  =  { A ,  B }  /\  A  e.  X )  ->  (
( V  =  { A }  \/  V  =  { A ,  B } )  /\  A  e.  X ) )
87ancomd 452 . . . . . . . 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 435 . . . . . 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 231 . . . . 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 4137 . . . . . . . 8  |-  ( -.  ( B  e.  _V  /\  B  =/=  A )  ->  { C ,  A ,  B }  =  { C ,  A } )
13 tprot 4092 . . . . . . . . . . . . 13  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
1413eqeq1i 2429 . . . . . . . . . . . 12  |-  ( { C ,  A ,  B }  =  { C ,  A }  <->  { A ,  B ,  C }  =  { C ,  A }
)
1514biimpi 197 . . . . . . . . . . 11  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { C ,  A }
)
16 prcom 4075 . . . . . . . . . . 11  |-  { C ,  A }  =  { A ,  C }
1715, 16syl6eq 2479 . . . . . . . . . 10  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  { A ,  B ,  C }  =  { A ,  C }
)
1817eqeq2d 2436 . . . . . . . . 9  |-  ( { C ,  A ,  B }  =  { C ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  C }
) )
19 olc 385 . . . . . . . . . . 11  |-  ( V  =  { A ,  C }  ->  ( V  =  { A }  \/  V  =  { A ,  C }
) )
20 1to2vfriswmgra 25720 . . . . . . . . . . 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 476 . . . . . . . . . 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 436 . . . . . . . . 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 231 . . . . . . . 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 4137 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  { B ,  A ,  C }  =  { B ,  A } )
27 tpcoma 4093 . . . . . . . . . . . . 13  |-  { B ,  A ,  C }  =  { A ,  B ,  C }
2827eqeq1i 2429 . . . . . . . . . . . 12  |-  ( { B ,  A ,  C }  =  { B ,  A }  <->  { A ,  B ,  C }  =  { B ,  A }
)
2928biimpi 197 . . . . . . . . . . 11  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { B ,  A }
)
30 prcom 4075 . . . . . . . . . . 11  |-  { B ,  A }  =  { A ,  B }
3129, 30syl6eq 2479 . . . . . . . . . 10  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  { A ,  B ,  C }  =  { A ,  B }
)
3231eqeq2d 2436 . . . . . . . . 9  |-  ( { B ,  A ,  C }  =  { B ,  A }  ->  ( V  =  { A ,  B ,  C }  <->  V  =  { A ,  B }
) )
336, 2sylan2 476 . . . . . . . . . . 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 436 . . . . . . . . . 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 231 . . . . . . . 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 458 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  e.  _V )
40 simpl 458 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  e.  _V )
4139, 40anim12i 568 . . . . . . . . . . . 12  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( B  e.  _V  /\  C  e.  _V ) )
4241ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  /\  V  =  { A ,  B ,  C }
)  ->  ( B  e.  _V  /\  C  e. 
_V ) )
4342anim1i 570 . . . . . . . . . 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 452 . . . . . . . . 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 986 . . . . . . . . 9  |-  ( ( A  e.  X  /\  B  e.  _V  /\  C  e.  _V )  <->  ( A  e.  X  /\  ( B  e.  _V  /\  C  e.  _V ) ) )
4644, 45sylibr 215 . . . . . . . 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 462 . . . . . . . . . . . . 13  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  B  =/=  A )
4847necomd 2695 . . . . . . . . . . . 12  |-  ( ( B  e.  _V  /\  B  =/=  A )  ->  A  =/=  B )
49 simpr 462 . . . . . . . . . . . . 13  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  C  =/=  A )
5049necomd 2695 . . . . . . . . . . . 12  |-  ( ( C  e.  _V  /\  C  =/=  A )  ->  A  =/=  C )
5148, 50anim12i 568 . . . . . . . . . . 11  |-  ( ( ( B  e.  _V  /\  B  =/=  A )  /\  ( C  e. 
_V  /\  C  =/=  A ) )  ->  ( A  =/=  B  /\  A  =/=  C ) )
5251anim1i 570 . . . . . . . . . 10  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( ( A  =/= 
B  /\  A  =/=  C )  /\  B  =/= 
C ) )
53 df-3an 984 . . . . . . . . . 10  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  <->  ( ( A  =/=  B  /\  A  =/=  C )  /\  B  =/=  C ) )
5452, 53sylibr 215 . . . . . . . . 9  |-  ( ( ( ( B  e. 
_V  /\  B  =/=  A )  /\  ( C  e.  _V  /\  C  =/=  A ) )  /\  B  =/=  C )  -> 
( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C
) )
5554ad2antrr 730 . . . . . . . 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 760 . . . . . . . 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 25719 . . . . . . . 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 1264 . . . . . . 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 613 . . . . . 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 950 . . . . 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 2737 . . . 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 380 . . 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 198 . 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 431 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 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   E!wreu 2777   _Vcvv 3081    \ cdif 3433   {csn 3996   {cpr 3998   {ctp 4000   class class class wbr 4420   ran crn 4851   FriendGrph cfrgra 25702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-hash 12516  df-usgra 25047  df-frgra 25703
This theorem is referenced by:  1to3vfriendship  25722
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