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Theorem 1vwmgra 24667
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 3927 . . . 4  |-  A. v  e.  (/)  ( { v ,  A }  e.  ran  E  /\  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E )
2 sneq 4032 . . . . . . . 8  |-  ( h  =  A  ->  { h }  =  { A } )
32difeq2d 3617 . . . . . . 7  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  ( { A }  \  { A } ) )
4 difid 3890 . . . . . . 7  |-  ( { A }  \  { A } )  =  (/)
53, 4syl6eq 2519 . . . . . 6  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  (/) )
6 preq2 4102 . . . . . . . 8  |-  ( h  =  A  ->  { v ,  h }  =  { v ,  A } )
76eleq1d 2531 . . . . . . 7  |-  ( h  =  A  ->  ( { v ,  h }  e.  ran  E  <->  { v ,  A }  e.  ran  E ) )
8 reueq1 3055 . . . . . . . 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 710 . . . . . 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 3067 . . . . 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 4058 . . . 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 233 . . 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 465 . 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 3610 . . . . 5  |-  ( V  =  { A }  ->  ( V  \  {
h } )  =  ( { A }  \  { h } ) )
16 reueq1 3055 . . . . . . 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 703 . . . . 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 3067 . . . 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 3062 . . 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 466 . 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 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   E!wreu 2811    \ cdif 3468   (/)c0 3780   {csn 4022   {cpr 4024   ran crn 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-sn 4023  df-pr 4025
This theorem is referenced by:  1to2vfriswmgra  24670
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