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 | stdpc4 | |- ( A. x -. ph -> [ y / x ] -. ph ) |
|
| 2 | sbn1 | |- ( [ y / x ] -. ph -> -. [ y / x ] ph ) |
|
| 3 | 1 2 | syl | |- ( A. x -. ph -> -. [ y / x ] ph ) |
| 4 | df-clab | |- ( y e. { x | ph } <-> [ y / x ] ph ) |
|
| 5 | 3 4 | sylnibr | |- ( A. x -. ph -> -. y e. { x | ph } ) |
| 6 | 5 | eq0rdv | |- ( A. x -. ph -> { x | ph } = (/) ) |