# Filter is Ideal in Dual Ordered Set

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## Theorem

Let $P = \left({S, \preceq}\right)$ be an ordered set.

Let $X$ be a subset of $S$.

Then

- $X$ is filter in $P$

- $X$ is ideal in $P^{-1}$

where $P^{-1} = \left({S, \succeq}\right)$ denotes the dual of $P$.

## Proof

- dual of $P^{-1}$ is $P$.

Hence by Ideal is Filter in Dual Ordered Set:

- the result fallows.

$\blacksquare$

## Sources

- Mizar article YELLOW_7:29

- Mizar article YELLOW_7:27