Metamath Proof Explorer


Theorem clwwlknondisj

Description: The sets of closed walks on different vertices are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018) (Revised by AV, 28-May-2021) (Revised by AV, 3-Mar-2022) (Proof shortened by AV, 28-Mar-2022)

Ref Expression
Assertion clwwlknondisj Disj x V x ClWWalksNOn G N

Proof

Step Hyp Ref Expression
1 clwwlknon x ClWWalksNOn G N = w N ClWWalksN G | w 0 = x
2 clwwlknon y ClWWalksNOn G N = w N ClWWalksN G | w 0 = y
3 1 2 ineq12i x ClWWalksNOn G N y ClWWalksNOn G N = w N ClWWalksN G | w 0 = x w N ClWWalksN G | w 0 = y
4 inrab w N ClWWalksN G | w 0 = x w N ClWWalksN G | w 0 = y = w N ClWWalksN G | w 0 = x w 0 = y
5 eqtr2 w 0 = x w 0 = y x = y
6 5 con3i ¬ x = y ¬ w 0 = x w 0 = y
7 6 ralrimivw ¬ x = y w N ClWWalksN G ¬ w 0 = x w 0 = y
8 rabeq0 w N ClWWalksN G | w 0 = x w 0 = y = w N ClWWalksN G ¬ w 0 = x w 0 = y
9 7 8 sylibr ¬ x = y w N ClWWalksN G | w 0 = x w 0 = y =
10 4 9 syl5eq ¬ x = y w N ClWWalksN G | w 0 = x w N ClWWalksN G | w 0 = y =
11 3 10 syl5eq ¬ x = y x ClWWalksNOn G N y ClWWalksNOn G N =
12 11 orri x = y x ClWWalksNOn G N y ClWWalksNOn G N =
13 12 rgen2w x V y V x = y x ClWWalksNOn G N y ClWWalksNOn G N =
14 oveq1 x = y x ClWWalksNOn G N = y ClWWalksNOn G N
15 14 disjor Disj x V x ClWWalksNOn G N x V y V x = y x ClWWalksNOn G N y ClWWalksNOn G N =
16 13 15 mpbir Disj x V x ClWWalksNOn G N