Description: A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 19-Jan-2021) (Revised by AV, 30-Oct-2021)
Ref | Expression | ||
---|---|---|---|
Hypothesis | 0pth.v | |
|
Assertion | 0pth | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0pth.v | |
|
2 | ispth | |
|
3 | 2 | a1i | |
4 | 3anass | |
|
5 | 4 | a1i | |
6 | funcnv0 | |
|
7 | hash0 | |
|
8 | 0le1 | |
|
9 | 7 8 | eqbrtri | |
10 | 1z | |
|
11 | 0z | |
|
12 | 7 11 | eqeltri | |
13 | fzon | |
|
14 | 10 12 13 | mp2an | |
15 | 9 14 | mpbi | |
16 | 15 | reseq2i | |
17 | res0 | |
|
18 | 16 17 | eqtri | |
19 | 18 | cnveqi | |
20 | 19 | funeqi | |
21 | 6 20 | mpbir | |
22 | 15 | imaeq2i | |
23 | ima0 | |
|
24 | 22 23 | eqtri | |
25 | 24 | ineq2i | |
26 | in0 | |
|
27 | 25 26 | eqtri | |
28 | 21 27 | pm3.2i | |
29 | 28 | biantru | |
30 | 5 29 | bitr4di | |
31 | 1 | 0trl | |
32 | 3 30 31 | 3bitrd | |