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Theorem numclwwlkdisj 25794
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 3745 . . . . 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 2449 . . . . . . . 8  |-  ( ( ( w `  0
)  =  x  /\  ( w `  0
)  =  y )  ->  x  =  y )
32con3i 140 . . . . . . 7  |-  ( -.  x  =  y  ->  -.  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) )
43ralrimivw 2840 . . . . . 6  |-  ( -.  x  =  y  ->  A. w  e.  ( C `  N )  -.  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) )
5 rabeq0 3784 . . . . . 6  |-  ( { w  e.  ( C `
 N )  |  ( ( w ` 
0 )  =  x  /\  ( w ` 
0 )  =  y ) }  =  (/)  <->  A. w  e.  ( C `  N )  -.  (
( w `  0
)  =  x  /\  ( w `  0
)  =  y ) )
64, 5sylibr 215 . . . . 5  |-  ( -.  x  =  y  ->  { w  e.  ( C `  N )  |  ( ( w `
 0 )  =  x  /\  ( w `
 0 )  =  y ) }  =  (/) )
71, 6syl5eq 2475 . . . 4  |-  ( -.  x  =  y  -> 
( { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }  i^i  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  y } )  =  (/) )
87orri 377 . . 3  |-  ( x  =  y  \/  ( { w  e.  ( C `  N )  |  ( w ` 
0 )  =  x }  i^i  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  y } )  =  (/) )
98rgen2w 2787 . 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 2437 . . . 4  |-  ( x  =  y  ->  (
( w `  0
)  =  x  <->  ( w `  0 )  =  y ) )
1110rabbidv 3072 . . 3  |-  ( x  =  y  ->  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }  =  {
w  e.  ( C `
 N )  |  ( w `  0
)  =  y } )
1211disjor 4405 . 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 212 1  |- Disj  x  e.  V  { w  e.  ( C `  N
)  |  ( w `
 0 )  =  x }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    /\ wa 370    = wceq 1437   A.wral 2775   {crab 2779    i^i cin 3435   (/)c0 3761  Disj wdisj 4391    |-> cmpt 4479   ` cfv 5598  (class class class)co 6302   0cc0 9540   NN0cn0 10870   ClWWalksN cclwwlkn 25463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rmo 2783  df-rab 2784  df-v 3083  df-dif 3439  df-in 3443  df-nul 3762  df-disj 4392
This theorem is referenced by:  numclwwlk4  25824
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