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Theorem numclwwlkdisj 30616
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 3615 . . . . 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 2455 . . . . . . . 8  |-  ( ( ( w `  0
)  =  x  /\  ( w `  0
)  =  y )  ->  x  =  y )
32con3i 135 . . . . . . 7  |-  ( -.  x  =  y  ->  -.  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) )
43ralrimivw 2794 . . . . . 6  |-  ( -.  x  =  y  ->  A. w  e.  ( C `  N )  -.  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) )
5 rabeq0 3652 . . . . . 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 2481 . . . 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 2778 . 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 2446 . . . 4  |-  ( x  =  y  ->  (
( w `  0
)  =  x  <->  ( w `  0 )  =  y ) )
1110rabbidv 2958 . . 3  |-  ( x  =  y  ->  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }  =  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  y } )
1211disjor 4269 . 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 1369   A.wral 2709   {crab 2713    i^i cin 3320   (/)c0 3630  Disj wdisj 4255    e. cmpt 4343   ` cfv 5411  (class class class)co 6086   0cc0 9274   NN0cn0 10571   ClWWalksN cclwwlkn 30357
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 2418
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-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rmo 2717  df-rab 2718  df-v 2968  df-dif 3324  df-in 3328  df-nul 3631  df-disj 4256
This theorem is referenced by:  numclwwlk4  30646
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