Theorem clwwlksndisj 26973

Description: The sets of closed walks starting at different vertices are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.)

Assertion

Ref	Expression
clwwlksndisj	|- Disj x e. V { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x }

Distinct variable groups:   x, G   x, N   x, V   x, w

Allowed substitution hints:   G( w)   N( w)   V( w)

Proof of Theorem clwwlksndisj

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inrab 3899	. . . . 5 |- ( { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x } i^i { w e. ( N ClWWalksN G ) | ( w ` 0 ) = y } ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = x /\ ( w ` 0 ) = y ) }
2		eqtr2 2642	. . . . . . . 8 |- ( ( ( w ` 0 ) = x /\ ( w ` 0 ) = y ) -> x = y )
3	2	con3i 150	. . . . . . 7 |- ( -. x = y -> -. ( ( w ` 0 ) = x /\ ( w ` 0 ) = y ) )
4	3	ralrimivw 2967	. . . . . 6 |- ( -. x = y -> A. w e. ( N ClWWalksN G ) -. ( ( w ` 0 ) = x /\ ( w ` 0 ) = y ) )
5		rabeq0 3957	. . . . . 6 |- ( { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = x /\ ( w ` 0 ) = y ) } = (/) <-> A. w e. ( N ClWWalksN G ) -. ( ( w ` 0 ) = x /\ ( w ` 0 ) = y ) )
6	4, 5	sylibr 224	. . . . 5 |- ( -. x = y -> { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = x /\ ( w ` 0 ) = y ) } = (/) )
7	1, 6	syl5eq 2668	. . . 4 |- ( -. x = y -> ( { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x } i^i { w e. ( N ClWWalksN G ) | ( w ` 0 ) = y } ) = (/) )
8	7	orri 391	. . 3 |- ( x = y \/ ( { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x } i^i { w e. ( N ClWWalksN G ) | ( w ` 0 ) = y } ) = (/) )
9	8	rgen2w 2925	. 2 |- A. x e. V A. y e. V ( x = y \/ ( { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x } i^i { w e. ( N ClWWalksN G ) | ( w ` 0 ) = y } ) = (/) )
10		eqeq2 2633	. . . 4 |- ( x = y -> ( ( w ` 0 ) = x <-> ( w ` 0 ) = y ) )
11	10	rabbidv 3189	. . 3 |- ( x = y -> { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x } = { w e. ( N ClWWalksN G ) | ( w ` 0 ) = y } )
12	11	disjor 4634	. 2 |- (Disj x e. V { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x } <-> A. x e. V A. y e. V ( x = y \/ ( { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x } i^i { w e. ( N ClWWalksN G ) | ( w ` 0 ) = y } ) = (/) ) )
13	9, 12	mpbir 221	1 |- Disj x e. V { w e. ( N ClWWalksN G ) | ( w ` 0 ) = x }

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   = wceq 1483   A.wral 2912   {crab 2916   i^i cin 3573   (/)c0 3915  Disj wdisj 4620   ` cfv 5888  (class class class)co 6650   0cc0 9936   ClWWalksN cclwwlksn 26876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  numclwwlk4  27244