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Theorem 1to2vfriswmgra 25732
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 25729 . . . . 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 26 . . . 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 436 . . 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 4426 . . . . . . . . 9  |-  ( V  =  { A ,  B }  ->  ( V FriendGrph  E 
<->  { A ,  B } FriendGrph  E ) )
54adantr 466 . . . . . . . 8  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
) )  ->  ( V FriendGrph  E  <->  { A ,  B } FriendGrph  E ) )
6 pm3.22 450 . . . . . . . . . . . 12  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  ( A  e.  X  /\  ( B  e.  _V  /\  A  =/=  B ) ) )
7 anass 653 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
)  <->  ( A  e.  X  /\  ( B  e.  _V  /\  A  =/=  B ) ) )
86, 7sylibr 215 . . . . . . . . . . 11  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
) )
9 frgra2v 25725 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  _V )  /\  A  =/=  B
)  ->  -.  { A ,  B } FriendGrph  E )
108, 9syl 17 . . . . . . . . . 10  |-  ( ( ( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
)  ->  -.  { A ,  B } FriendGrph  E )
1110adantl 467 . . . . . . . . 9  |-  ( ( V  =  { A ,  B }  /\  (
( B  e.  _V  /\  A  =/=  B )  /\  A  e.  X
) )  ->  -.  { A ,  B } FriendGrph  E )
1211pm2.21d 109 . . . . . . . 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 218 . . . . . . 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 436 . . . . . 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 435 . . . . 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 81 . . . 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 490 . . . . . . 7  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  <-> 
( -.  B  e. 
_V  \/  -.  A  =/=  B ) )
18 prprc2 4111 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { A ,  B }  =  { A } )
19 nne 2620 . . . . . . . . 9  |-  ( -.  A  =/=  B  <->  A  =  B )
20 preq2 4080 . . . . . . . . . . 11  |-  ( B  =  A  ->  { A ,  B }  =  { A ,  A }
)
2120eqcoms 2434 . . . . . . . . . 10  |-  ( A  =  B  ->  { A ,  B }  =  { A ,  A }
)
22 dfsn2 4011 . . . . . . . . . 10  |-  { A }  =  { A ,  A }
2321, 22syl6eqr 2481 . . . . . . . . 9  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2419, 23sylbi 198 . . . . . . . 8  |-  ( -.  A  =/=  B  ->  { A ,  B }  =  { A } )
2518, 24jaoi 380 . . . . . . 7  |-  ( ( -.  B  e.  _V  \/  -.  A  =/=  B
)  ->  { A ,  B }  =  { A } )
2617, 25sylbi 198 . . . . . 6  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  ->  { A ,  B }  =  { A } )
2726eqeq2d 2436 . . . . 5  |-  ( -.  ( B  e.  _V  /\  A  =/=  B )  ->  ( V  =  { A ,  B } 
<->  V  =  { A } ) )
2827, 3syl6bi 231 . . . 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 167 . . 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 380 . 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 431 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   E!wreu 2773   _Vcvv 3080    \ cdif 3433   {csn 3998   {cpr 4000   class class class wbr 4423   ran crn 4854   FriendGrph cfrgra 25714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  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 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12522  df-usgra 25058  df-frgra 25715
This theorem is referenced by:  1to3vfriswmgra  25733
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