Consistency of intersection and union of cuts

Hypothesis: If $C_1$ and $C_2$ are two consistent cuts, then $C_1\cap C_2$ is a consistent cut.

Proof: We know that if

$$e\to e’\wedge e’\in C_1\cap C_2$$

then since $e\to e^{\prime }$, $e$ will be in every cut that includes $e^{\prime }$, so

$$e\in C_1\wedge e\in C_2\Rightarrow e\in C_1\cap C_2$$

Hypothesis: If $C_1$ and $C_2$ are two consistent cuts, then $C_1\cup C_2$ is a consistent cut.

Proof: Similarly to the proof for the intersection, we know that if

$$e\to e’\wedge e’\in C_1\cup C_2$$

then since $e\to e^{\prime }$, $e$ will be in every cut that includes $e^{\prime }$, so

$$e\in C_1\vee e\in C_2\Rightarrow e\in C_1\cup C_2$$