MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1vwmgra Structured version   Unicode version

Theorem 1vwmgra 25301
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 3877 . . . 4  |-  A. v  e.  (/)  ( { v ,  A }  e.  ran  E  /\  E! w  e.  ( { A }  \  { A } ) { v ,  w }  e.  ran  E )
2 sneq 3981 . . . . . . . 8  |-  ( h  =  A  ->  { h }  =  { A } )
32difeq2d 3560 . . . . . . 7  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  ( { A }  \  { A } ) )
4 difid 3839 . . . . . . 7  |-  ( { A }  \  { A } )  =  (/)
53, 4syl6eq 2459 . . . . . 6  |-  ( h  =  A  ->  ( { A }  \  {
h } )  =  (/) )
6 preq2 4051 . . . . . . . 8  |-  ( h  =  A  ->  { v ,  h }  =  { v ,  A } )
76eleq1d 2471 . . . . . . 7  |-  ( h  =  A  ->  ( { v ,  h }  e.  ran  E  <->  { v ,  A }  e.  ran  E ) )
8 reueq1 3005 . . . . . . . 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 17 . . . . . . 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 709 . . . . . 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 3017 . . . . 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 4007 . . . 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 463 . 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 3553 . . . . 5  |-  ( V  =  { A }  ->  ( V  \  {
h } )  =  ( { A }  \  { h } ) )
16 reueq1 3005 . . . . . . 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 17 . . . . . 6  |-  ( V  =  { A }  ->  ( E! w  e.  ( V  \  {
h } ) { v ,  w }  e.  ran  E  <->  E! w  e.  ( { A }  \  { h } ) { v ,  w }  e.  ran  E ) )
1817anbi2d 702 . . . . 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 3017 . . . 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 3012 . . 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 464 . 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   E.wrex 2754   E!wreu 2755    \ cdif 3410   (/)c0 3737   {csn 3971   {cpr 3973   ran crn 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-sn 3972  df-pr 3974
This theorem is referenced by:  1to2vfriswmgra  25304
  Copyright terms: Public domain W3C validator