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Theorem 1to2vfriswmgra 24679
Description: Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
1to2vfriswmgra  |-  ( ( 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 ) ) )
Distinct variable groups:    A, h, v, w    B, h, v, w    h, E, v, w    h, V, v, w    v, X, w
Allowed substitution hint:    X( h)

Proof of Theorem 1to2vfriswmgra
StepHypRef Expression
1 1vwmgra 24676 . . . . 5  |-  ( ( A  e.  X  /\  V  =  { A } )  ->  E. h  e.  V  A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E ) )
21a1d 25 . . . 4  |-  ( ( A  e.  X  /\  V  =  { A } )  ->  ( 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 . . 3  |-  ( V  =  { A }  ->  ( 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 breq1 4450 . . . . . . . . 9  |-  ( V  =  { A ,  B }  ->  ( V FriendGrph  E 
<->  { A ,  B } FriendGrph  E ) )
54adantr 465 . . . . . . . 8  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
) )  ->  ( V FriendGrph  E  <->  { A ,  B } FriendGrph  E ) )
6 pm3.22 449 . . . . . . . . . . . 12  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  ( A  e.  X  /\  ( B  e.  _V  /\  A  =/=  B ) ) )
7 anass 649 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
)  <->  ( A  e.  X  /\  ( B  e.  _V  /\  A  =/=  B ) ) )
86, 7sylibr 212 . . . . . . . . . . 11  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
) )
9 frgra2v 24672 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
)  ->  -.  { A ,  B } FriendGrph  E )
108, 9syl 16 . . . . . . . . . 10  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  -.  { A ,  B } FriendGrph  E )
1110adantl 466 . . . . . . . . 9  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
) )  ->  -.  { A ,  B } FriendGrph  E )
1211pm2.21d 106 . . . . . . . 8  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
) )  ->  ( { A ,  B } 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 ) ) )
135, 12sylbid 215 . . . . . . 7  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _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 ) ) )
1413expcom 435 . . . . . 6  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  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 ) ) ) )
1514ex 434 . . . . 5  |-  ( ( B  e.  _V  /\  A  =/=  B )  -> 
( 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 ) ) ) ) )
1615com23 78 . . . 4  |-  ( ( B  e.  _V  /\  A  =/=  B )  -> 
( 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 ) ) ) ) )
17 ianor 488 . . . . . . 7  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  <-> 
( -.  B  e. 
_V  \/  -.  A  =/=  B ) )
18 prprc2 4138 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { A ,  B }  =  { A } )
19 nne 2668 . . . . . . . . 9  |-  ( -.  A  =/=  B  <->  A  =  B )
20 preq2 4107 . . . . . . . . . . 11  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
2120eqcoms 2479 . . . . . . . . . 10  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
22 dfsn2 4040 . . . . . . . . . 10  |-  { A }  =  { A ,  A }
2321, 22syl6eqr 2526 . . . . . . . . 9  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2419, 23sylbi 195 . . . . . . . 8  |-  ( -.  A  =/=  B  ->  { A ,  B }  =  { A } )
2518, 24jaoi 379 . . . . . . 7  |-  ( ( -.  B  e.  _V  \/  -.  A  =/=  B
)  ->  { A ,  B }  =  { A } )
2617, 25sylbi 195 . . . . . 6  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  ->  { A ,  B }  =  { A } )
2726eqeq2d 2481 . . . . 5  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  ->  ( V  =  { A ,  B } 
<->  V  =  { A } ) )
2827, 3syl6bi 228 . . . 4  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  ->  ( 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 ) ) ) ) )
2916, 28pm2.61i 164 . . 3  |-  ( 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 ) ) ) )
303, 29jaoi 379 . 2  |-  ( ( 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 ) ) ) )
3130impcom 430 1  |-  ( ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816   _Vcvv 3113    \ cdif 3473   {csn 4027   {cpr 4029   class class class wbr 4447   ran crn 5000   FriendGrph cfrgra 24661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12368  df-usgra 24006  df-frgra 24662
This theorem is referenced by:  1to3vfriswmgra  24680
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