The question is:
The set $S$ is defined as
- $\varnothing \in S$,
- If $x \in S$, then also $\{x\} \cup x \in S$.
Prove or disprove it is partial ordering.
So the set $S$ looks like $$\{\varnothing, \{\varnothing\}\cup\varnothing, \{\{\varnothing\}\cup\varnothing\}\cup\{\varnothing\}\cup\varnothing,\cdots \},$$ right?
And I think it is reflexive because every set is a subset of itself, antisymmetric because two sets are subset of each other iff they are same, and transitive because $\subseteq$ is transitive. So it is partial ordering.
I don't know how to prove this formally, that is, use symbols and formal language to prove it.