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Theorem 1vwmgra 28427
Description: Every graph with one vertex is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
1vwmgra  |-  ( ( 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 ) )
Distinct variable groups:    A, h, v, w    h, E    h, V, v, w
Allowed substitution hints:    E( w, v)    X( w, v, h)

Proof of Theorem 1vwmgra
StepHypRef Expression
1 ral0 3571 . . . 4  |-  A. v  e.  (/)  ( { v ,  A }  e.  ran  E  /\  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E )
2 sneq 3664 . . . . . . . 8  |-  ( h  =  A  ->  { h }  =  { A } )
32difeq2d 3307 . . . . . . 7  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  ( { A }  \  { A } ) )
4 difid 3535 . . . . . . 7  |-  ( { A }  \  { A } )  =  (/)
53, 4syl6eq 2344 . . . . . 6  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  (/) )
6 preq2 3720 . . . . . . . 8  |-  ( h  =  A  ->  { v ,  h }  =  { v ,  A } )
76eleq1d 2362 . . . . . . 7  |-  ( h  =  A  ->  ( { v ,  h }  e.  ran  E  <->  { v ,  A }  e.  ran  E ) )
8 reueq1 2751 . . . . . . . 8  |-  ( ( { A }  \  { h } )  =  ( { A }  \  { A }
)  ->  ( E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E  <->  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E ) )
93, 8syl 15 . . . . . . 7  |-  ( h  =  A  ->  ( E! w  e.  ( { A }  \  {
h } ) { v ,  w }  e.  ran  E  <->  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E ) )
107, 9anbi12d 691 . . . . . 6  |-  ( h  =  A  ->  (
( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E )  <-> 
( { v ,  A }  e.  ran  E  /\  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E ) ) )
115, 10raleqbidv 2761 . . . . 5  |-  ( h  =  A  ->  ( A. v  e.  ( { A }  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  {
h } ) { v ,  w }  e.  ran  E )  <->  A. v  e.  (/)  ( { v ,  A }  e.  ran  E  /\  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E ) ) )
1211rexsng 3686 . . . 4  |-  ( A  e.  X  ->  ( E. h  e.  { A } A. v  e.  ( { A }  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E )  <->  A. v  e.  (/)  ( { v ,  A }  e.  ran  E  /\  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E ) ) )
131, 12mpbiri 224 . . 3  |-  ( A  e.  X  ->  E. h  e.  { A } A. v  e.  ( { A }  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) )
1413adantr 451 . 2  |-  ( ( A  e.  X  /\  V  =  { A } )  ->  E. h  e.  { A } A. v  e.  ( { A }  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) )
15 id 19 . . . 4  |-  ( V  =  { A }  ->  V  =  { A } )
16 difeq1 3300 . . . . 5  |-  ( V  =  { A }  ->  ( V  \  {
h } )  =  ( { A }  \  { h } ) )
17 reueq1 2751 . . . . . . 7  |-  ( ( V  \  { h } )  =  ( { A }  \  { h } )  ->  ( E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E  <->  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) )
1816, 17syl 15 . . . . . 6  |-  ( V  =  { A }  ->  ( E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E  <->  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) )
1918anbi2d 684 . . . . 5  |-  ( V  =  { A }  ->  ( ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E )  <->  ( {
v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) ) )
2016, 19raleqbidv 2761 . . . 4  |-  ( V  =  { A }  ->  ( A. v  e.  ( V  \  {
h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h }
) { v ,  w }  e.  ran  E )  <->  A. v  e.  ( { A }  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) ) )
2115, 20rexeqbidv 2762 . . 3  |-  ( 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 )  <->  E. h  e.  { A } A. v  e.  ( { A }  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) ) )
2221adantl 452 . 2  |-  ( ( 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 )  <->  E. h  e.  { A } A. v  e.  ( { A }  \  { h } ) ( { v ,  h }  e.  ran  E  /\  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) ) )
2314, 22mpbird 223 1  |-  ( ( 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 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558    \ cdif 3162   (/)c0 3468   {csn 3653   {cpr 3654   ran crn 4706
This theorem is referenced by:  1to2vfriswmgra  28430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660
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