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Theorem 1to2vfriswmgra 28323
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 28320 . . . . 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 23 . . . 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 425 . . 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 4207 . . . . . . . . 9  |-  ( V  =  { A ,  B }  ->  ( V FriendGrph  E 
<->  { A ,  B } FriendGrph  E ) )
54adantr 452 . . . . . . . 8  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
) )  ->  ( V FriendGrph  E  <->  { A ,  B } FriendGrph  E ) )
6 pm3.22 437 . . . . . . . . . . . 12  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  ( A  e.  X  /\  ( B  e.  _V  /\  A  =/=  B ) ) )
7 anass 631 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
)  <->  ( A  e.  X  /\  ( B  e.  _V  /\  A  =/=  B ) ) )
86, 7sylibr 204 . . . . . . . . . . 11  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
) )
9 frgra2v 28316 . . . . . . . . . . 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 453 . . . . . . . . 9  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
) )  ->  -.  { A ,  B } FriendGrph  E )
1211pm2.21d 100 . . . . . . . 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 207 . . . . . . 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 425 . . . . . 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 424 . . . . 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 74 . . . 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 475 . . . . . . 7  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  <-> 
( -.  B  e. 
_V  \/  -.  A  =/=  B ) )
18 prprc2 3907 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { A ,  B }  =  { A } )
19 nne 2602 . . . . . . . . 9  |-  ( -.  A  =/=  B  <->  A  =  B )
20 preq2 3876 . . . . . . . . . . 11  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
2120eqcoms 2438 . . . . . . . . . 10  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
22 dfsn2 3820 . . . . . . . . . 10  |-  { A }  =  { A ,  A }
2321, 22syl6eqr 2485 . . . . . . . . 9  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2419, 23sylbi 188 . . . . . . . 8  |-  ( -.  A  =/=  B  ->  { A ,  B }  =  { A } )
2518, 24jaoi 369 . . . . . . 7  |-  ( ( -.  B  e.  _V  \/  -.  A  =/=  B
)  ->  { A ,  B }  =  { A } )
2617, 25sylbi 188 . . . . . 6  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  ->  { A ,  B }  =  { A } )
2726eqeq2d 2446 . . . . 5  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  ->  ( V  =  { A ,  B } 
<->  V  =  { A } ) )
2827, 3syl6bi 220 . . . 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 158 . . 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 369 . 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 420 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   E!wreu 2699   _Vcvv 2948    \ cdif 3309   {csn 3806   {cpr 3807   class class class wbr 4204   ran crn 4871   FriendGrph cfrgra 28305
This theorem is referenced by:  1to3vfriswmgra  28324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611  df-usgra 21359  df-frgra 28306
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