A proof was given reproduced here:
Prove: $A \cup \varnothing = A$
Let $a\in A\cup \varnothing$. Then $a\in A$ or $a\in\varnothing$. Since $a\in\varnothing$ is false regardless of $a$, but we assumed $a\in A\cup \varnothing$, it must be that $a\in A$ is true, so that $A\cup \varnothing \subseteq A$. Conversely, $A\cup \varnothing \supseteq A$ trivially, so $A=A\cup\varnothing$.
I never took a course on elementary logic (or real analysis for that matters) so it escapes me:
it must be that $a\in A$ is true $\implies A\cup \varnothing \subseteq A$
How does a truth statement (a sentence in Englisch) just translates into a set inclusion??
Should there be something in between:
it must be that $a\in A$ is true [and in logic, "true" relates to set inclusion like this] $ A\cup \varnothing \subseteq A$
Can someone bridge this gap?