Let $ \succ $ be a binary relationship on the set $X$ such that, given $ x, y, z\in X $, if $x\succ y$:

1. (Asymmetry): $\neg(y\succ x)$,
2. (Negative transitivity): $(x\succ z) \vee (z\succ y)$.

Let us define the abbreviations:

3. $x\succeq y \;:=\; \neg(y\succ x) $,

4. $x \sim y \;:=\; x\succeq y\; \wedge \;y \succeq x$.

As usual, the relations $\succ, \succeq, \sim$ denote strong preference, weak preference, and indifference.

Intuition suggests that I can conclude: $$x\succeq y \; \leftrightarrow \;(x\succ y\; \vee \;x\sim y) $$

If so, how can I derive it formally? Any useful references?

Let $ \succ $ be a binary relationship on the set $X$ such that, given any $ x, y, z\in X $, if $x\succ y$:

1. (Asymmetry): $\neg(y\succ x)$,
2. (Negative transitivity): $(x\succ z) \vee (z\succ y)$.

Let us define the abbreviations:

3. $x\succeq y \;:=\; \neg(y\succ x) $,

4. $x \sim y \;:=\; x\succeq y\; \wedge \;y \succeq x$.

As usual, the relations $\succ, \succeq, \sim$ denote strong preference, weak preference, and indifference.

Intuition suggests that I can conclude: $$x\succeq y \; \leftrightarrow \;(x\succ y\; \vee \;x\sim y) $$

If so, how can I derive it formally? Any useful references?