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Theorem 1vwmgra 28294
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 3724 . . . 4  |-  A. v  e.  (/)  ( { v ,  A }  e.  ran  E  /\  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E )
2 sneq 3817 . . . . . . . 8  |-  ( h  =  A  ->  { h }  =  { A } )
32difeq2d 3457 . . . . . . 7  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  ( { A }  \  { A } ) )
4 difid 3688 . . . . . . 7  |-  ( { A }  \  { A } )  =  (/)
53, 4syl6eq 2483 . . . . . 6  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  (/) )
6 preq2 3876 . . . . . . . 8  |-  ( h  =  A  ->  { v ,  h }  =  { v ,  A } )
76eleq1d 2501 . . . . . . 7  |-  ( h  =  A  ->  ( { v ,  h }  e.  ran  E  <->  { v ,  A }  e.  ran  E ) )
8 reueq1 2898 . . . . . . . 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 16 . . . . . . 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 692 . . . . . 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 2908 . . . . 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 3839 . . . 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 225 . . 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 452 . 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 difeq1 3450 . . . . 5  |-  ( V  =  { A }  ->  ( V  \  {
h } )  =  ( { A }  \  { h } ) )
16 reueq1 2898 . . . . . . 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 ) )
1715, 16syl 16 . . . . . 6  |-  ( V  =  { A }  ->  ( E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E  <->  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) )
1817anbi2d 685 . . . . 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 ) ) )
1915, 18raleqbidv 2908 . . . 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 ) ) )
2019rexeqbi1dv 2905 . . 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 ) ) )
2120adantl 453 . 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 ) ) )
2214, 21mpbird 224 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   E!wreu 2699    \ cdif 3309   (/)c0 3620   {csn 3806   {cpr 3807   ran crn 4871
This theorem is referenced by:  1to2vfriswmgra  28297
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813
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