Description: Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997) (Proof shortened by Andrew Salmon, 25-Jul-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | po0 | |- R Po (/) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ral0 | |- A. x e. (/) A. y e. (/) A. z e. (/) ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) |
|
| 2 | df-po | |- ( R Po (/) <-> A. x e. (/) A. y e. (/) A. z e. (/) ( -. x R x /\ ( ( x R y /\ y R z ) -> x R z ) ) ) |
|
| 3 | 1 2 | mpbir | |- R Po (/) |