superfluity of the third defining property for finite consequence operator

In this entry, we demonstrate the claim made in section 1 of thehttp://planetmath.org/node/8646parent entry that the defining conditions forfinitary consequence operator given there are redundant becauseone of them may be derived from the other two.

Theorem.

Let $L$ be a set. Suppose that a mapping $C\mathrm{:}\mathrm{P}\mathit{}\mathrm{(}L\mathrm{)}\mathrm{\to }\mathrm{P}\mathit{}\mathrm{(}L\mathrm{)}$ satisfies the following three properties:

1. For all $X\subseteq L$, it happens that $X\subseteq C(X)$.

2. $C\circ C=C$

3. For all $X\in L$, it happens that $C(X)=\bigcup _{Y\in ℱ(X)}C(Y)$.

Then $C$ also satisfies the following property: For all $X\mathrm{,}Y\mathrm{\subseteq }L$, if $X\mathrm{\subseteq }Y$, then $C\mathit{}\mathrm{(}X\mathrm{)}\mathrm{\subseteq }C\mathit{}\mathrm{(}Y\mathrm{)}$.