Description: The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | left0s | |- ( _L ` 0s ) = (/) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno | |- 0s e. No |
|
2 | leftssold | |- ( 0s e. No -> ( _L ` 0s ) C_ ( _Old ` ( bday ` 0s ) ) ) |
|
3 | 1 2 | ax-mp | |- ( _L ` 0s ) C_ ( _Old ` ( bday ` 0s ) ) |
4 | bday0s | |- ( bday ` 0s ) = (/) |
|
5 | 4 | fveq2i | |- ( _Old ` ( bday ` 0s ) ) = ( _Old ` (/) ) |
6 | old0 | |- ( _Old ` (/) ) = (/) |
|
7 | 5 6 | eqtri | |- ( _Old ` ( bday ` 0s ) ) = (/) |
8 | sseq0 | |- ( ( ( _L ` 0s ) C_ ( _Old ` ( bday ` 0s ) ) /\ ( _Old ` ( bday ` 0s ) ) = (/) ) -> ( _L ` 0s ) = (/) ) |
|
9 | 3 7 8 | mp2an | |- ( _L ` 0s ) = (/) |