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Theorem numclwwlkdisj 24952
Description: The sets of closed walks starting at different vertices in an undirected simple graph are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
Hypothesis
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
Assertion
Ref Expression
numclwwlkdisj  |- Disj  x  e.  V  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }
Distinct variable groups:    n, E    n, N    n, V    w, C, x    x, E    w, N, x    x, V
Allowed substitution hints:    C( n)    E( w)    V( w)

Proof of Theorem numclwwlkdisj
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 inrab 3755 . . . . 5  |-  ( { w  e.  ( C `
 N )  |  ( w `  0
)  =  x }  i^i  { w  e.  ( C `  N )  |  ( w ` 
0 )  =  y } )  =  {
w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) }
2 eqtr2 2470 . . . . . . . 8  |-  ( ( ( w `  0
)  =  x  /\  ( w `  0
)  =  y )  ->  x  =  y )
32con3i 135 . . . . . . 7  |-  ( -.  x  =  y  ->  -.  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) )
43ralrimivw 2858 . . . . . 6  |-  ( -.  x  =  y  ->  A. w  e.  ( C `  N )  -.  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) )
5 rabeq0 3793 . . . . . 6  |-  ( { w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) }  =  (/)  <->  A. w  e.  ( C `  N )  -.  (
( w `  0
)  =  x  /\  ( w `  0
)  =  y ) )
64, 5sylibr 212 . . . . 5  |-  ( -.  x  =  y  ->  { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  x  /\  ( w `
 0 )  =  y ) }  =  (/) )
71, 6syl5eq 2496 . . . 4  |-  ( -.  x  =  y  -> 
( { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }  i^i  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  y } )  =  (/) )
87orri 376 . . 3  |-  ( x  =  y  \/  ( { w  e.  ( C `  N )  |  ( w ` 
0 )  =  x }  i^i  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  y } )  =  (/) )
98rgen2w 2805 . 2  |-  A. x  e.  V  A. y  e.  V  ( x  =  y  \/  ( { w  e.  ( C `  N )  |  ( w ` 
0 )  =  x }  i^i  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  y } )  =  (/) )
10 eqeq2 2458 . . . 4  |-  ( x  =  y  ->  (
( w `  0
)  =  x  <->  ( w `  0 )  =  y ) )
1110rabbidv 3087 . . 3  |-  ( x  =  y  ->  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }  =  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  y } )
1211disjor 4421 . 2  |-  (Disj  x  e.  V  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }  <->  A. x  e.  V  A. y  e.  V  ( x  =  y  \/  ( { w  e.  ( C `  N )  |  ( w ` 
0 )  =  x }  i^i  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  y } )  =  (/) ) )
139, 12mpbir 209 1  |- Disj  x  e.  V  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1383   A.wral 2793   {crab 2797    i^i cin 3460   (/)c0 3770  Disj wdisj 4407    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   0cc0 9495   NN0cn0 10801   ClWWalksN cclwwlkn 24621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rmo 2801  df-rab 2802  df-v 3097  df-dif 3464  df-in 3468  df-nul 3771  df-disj 4408
This theorem is referenced by:  numclwwlk4  24982
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