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Theorem clwlknclwlkdifs 30576
Description: The set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between the set of walks of length n starting with this vertex and the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
Hypotheses
Ref Expression
clwlknclwlkdif.a  |-  A  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( ( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X ) }
clwlknclwlkdif.b  |-  B  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( ( lastS  `  w )  =  ( w ` 
0 )  /\  (
w `  0 )  =  X ) }
Assertion
Ref Expression
clwlknclwlkdifs  |-  A  =  ( { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  X }  \  B )

Proof of Theorem clwlknclwlkdifs
StepHypRef Expression
1 clwlknclwlkdif.a . 2  |-  A  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( ( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X ) }
2 clwlknclwlkdif.b . . . 4  |-  B  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( ( lastS  `  w )  =  ( w ` 
0 )  /\  (
w `  0 )  =  X ) }
32difeq2i 3470 . . 3  |-  ( { w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  X }  \  B
)  =  ( { w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  X }  \  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( ( lastS  `  w )  =  ( w `  0 )  /\  ( w ` 
0 )  =  X ) } )
4 difrab 3623 . . 3  |-  ( { w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  X }  \  {
w  e.  ( ( V WWalksN  E ) `  N
)  |  ( ( lastS  `  w )  =  ( w `  0 )  /\  ( w ` 
0 )  =  X ) } )  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( ( w `  0
)  =  X  /\  -.  ( ( lastS  `  w
)  =  ( w `
 0 )  /\  ( w `  0
)  =  X ) ) }
5 ianor 488 . . . . . . . 8  |-  ( -.  ( ( lastS  `  w
)  =  ( w `
 0 )  /\  ( w `  0
)  =  X )  <-> 
( -.  ( lastS  `  w
)  =  ( w `
 0 )  \/ 
-.  ( w ` 
0 )  =  X ) )
6 eqeq2 2451 . . . . . . . . . . . 12  |-  ( ( w `  0 )  =  X  ->  (
( lastS  `  w )  =  ( w `  0
)  <->  ( lastS  `  w )  =  X ) )
76notbid 294 . . . . . . . . . . 11  |-  ( ( w `  0 )  =  X  ->  ( -.  ( lastS  `  w )  =  ( w ` 
0 )  <->  -.  ( lastS  `  w )  =  X ) )
8 df-ne 2607 . . . . . . . . . . . . . 14  |-  ( ( lastS  `  w )  =/=  X  <->  -.  ( lastS  `  w )  =  X )
98biimpri 206 . . . . . . . . . . . . 13  |-  ( -.  ( lastS  `  w )  =  X  ->  ( lastS  `  w
)  =/=  X )
109anim2i 569 . . . . . . . . . . . 12  |-  ( ( ( w `  0
)  =  X  /\  -.  ( lastS  `  w )  =  X )  -> 
( ( w ` 
0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) )
1110ex 434 . . . . . . . . . . 11  |-  ( ( w `  0 )  =  X  ->  ( -.  ( lastS  `  w )  =  X  ->  (
( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X ) ) )
127, 11sylbid 215 . . . . . . . . . 10  |-  ( ( w `  0 )  =  X  ->  ( -.  ( lastS  `  w )  =  ( w ` 
0 )  ->  (
( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X ) ) )
1312com12 31 . . . . . . . . 9  |-  ( -.  ( lastS  `  w )  =  ( w ` 
0 )  ->  (
( w `  0
)  =  X  -> 
( ( w ` 
0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) ) )
14 pm2.21 108 . . . . . . . . 9  |-  ( -.  ( w `  0
)  =  X  -> 
( ( w ` 
0 )  =  X  ->  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) ) )
1513, 14jaoi 379 . . . . . . . 8  |-  ( ( -.  ( lastS  `  w
)  =  ( w `
 0 )  \/ 
-.  ( w ` 
0 )  =  X )  ->  ( (
w `  0 )  =  X  ->  ( ( w `  0 )  =  X  /\  ( lastS  `  w )  =/=  X
) ) )
165, 15sylbi 195 . . . . . . 7  |-  ( -.  ( ( lastS  `  w
)  =  ( w `
 0 )  /\  ( w `  0
)  =  X )  ->  ( ( w `
 0 )  =  X  ->  ( (
w `  0 )  =  X  /\  ( lastS  `  w )  =/=  X
) ) )
1716impcom 430 . . . . . 6  |-  ( ( ( w `  0
)  =  X  /\  -.  ( ( lastS  `  w
)  =  ( w `
 0 )  /\  ( w `  0
)  =  X ) )  ->  ( (
w `  0 )  =  X  /\  ( lastS  `  w )  =/=  X
) )
18 simpl 457 . . . . . . 7  |-  ( ( ( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X )  ->  (
w `  0 )  =  X )
19 neeq2 2616 . . . . . . . . . . 11  |-  ( X  =  ( w ` 
0 )  ->  (
( lastS  `  w )  =/= 
X  <->  ( lastS  `  w )  =/=  ( w ` 
0 ) ) )
2019eqcoms 2445 . . . . . . . . . 10  |-  ( ( w `  0 )  =  X  ->  (
( lastS  `  w )  =/= 
X  <->  ( lastS  `  w )  =/=  ( w ` 
0 ) ) )
21 df-ne 2607 . . . . . . . . . . 11  |-  ( ( lastS  `  w )  =/=  (
w `  0 )  <->  -.  ( lastS  `  w )  =  ( w ` 
0 ) )
2221biimpi 194 . . . . . . . . . 10  |-  ( ( lastS  `  w )  =/=  (
w `  0 )  ->  -.  ( lastS  `  w
)  =  ( w `
 0 ) )
2320, 22syl6bi 228 . . . . . . . . 9  |-  ( ( w `  0 )  =  X  ->  (
( lastS  `  w )  =/= 
X  ->  -.  ( lastS  `  w )  =  ( w `  0 ) ) )
2423imp 429 . . . . . . . 8  |-  ( ( ( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X )  ->  -.  ( lastS  `  w )  =  ( w `  0
) )
2524intnanrd 908 . . . . . . 7  |-  ( ( ( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X )  ->  -.  ( ( lastS  `  w )  =  ( w ` 
0 )  /\  (
w `  0 )  =  X ) )
2618, 25jca 532 . . . . . 6  |-  ( ( ( w `  0
)  =  X  /\  ( lastS  `  w )  =/= 
X )  ->  (
( w `  0
)  =  X  /\  -.  ( ( lastS  `  w
)  =  ( w `
 0 )  /\  ( w `  0
)  =  X ) ) )
2717, 26impbii 188 . . . . 5  |-  ( ( ( w `  0
)  =  X  /\  -.  ( ( lastS  `  w
)  =  ( w `
 0 )  /\  ( w `  0
)  =  X ) )  <->  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) )
2827a1i 11 . . . 4  |-  ( w  e.  ( ( V WWalksN  E ) `  N
)  ->  ( (
( w `  0
)  =  X  /\  -.  ( ( lastS  `  w
)  =  ( w `
 0 )  /\  ( w `  0
)  =  X ) )  <->  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) ) )
2928rabbiia 2960 . . 3  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( ( w `
 0 )  =  X  /\  -.  (
( lastS  `  w )  =  ( w `  0
)  /\  ( w `  0 )  =  X ) ) }  =  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) }
303, 4, 293eqtrri 2467 . 2  |-  { w  e.  ( ( V WWalksN  E
) `  N )  |  ( ( w `
 0 )  =  X  /\  ( lastS  `  w
)  =/=  X ) }  =  ( { w  e.  ( ( V WWalksN  E ) `  N
)  |  ( w `
 0 )  =  X }  \  B
)
311, 30eqtri 2462 1  |-  A  =  ( { w  e.  ( ( V WWalksN  E
) `  N )  |  ( w ` 
0 )  =  X }  \  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   {crab 2718    \ cdif 3324   ` cfv 5417  (class class class)co 6090   0cc0 9281   lastS clsw 12221   WWalksN cwwlkn 30310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rab 2723  df-v 2973  df-dif 3330
This theorem is referenced by:  clwlknclwlkdifnum  30577
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