Description: The class of sets verifying a falsity is the empty set (closed form of abf ). (Contributed by BJ, 24-Jul-2019) (Proof modification is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-ab0 | |- ( A. x -. ph -> { x | ph } = (/) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-5 | |- ( A. x -. ph -> A. y A. x -. ph ) |
|
| 2 | stdpc4 | |- ( A. x -. ph -> [ y / x ] -. ph ) |
|
| 3 | sbn | |- ( [ y / x ] -. ph <-> -. [ y / x ] ph ) |
|
| 4 | 2 3 | sylib | |- ( A. x -. ph -> -. [ y / x ] ph ) |
| 5 | df-clab | |- ( y e. { x | ph } <-> [ y / x ] ph ) |
|
| 6 | 4 5 | sylnibr | |- ( A. x -. ph -> -. y e. { x | ph } ) |
| 7 | 1 6 | alrimih | |- ( A. x -. ph -> A. y -. y e. { x | ph } ) |
| 8 | eq0 | |- ( { x | ph } = (/) <-> A. y -. y e. { x | ph } ) |
|
| 9 | 7 8 | sylibr | |- ( A. x -. ph -> { x | ph } = (/) ) |