Proof:∅ ⊆ A

From CS2800 wiki
for any set [math]X [/math], [math]\href{/cs2800/wiki/index.php/%E2%88%85}{∅} \href{/cs2800/wiki/index.php/%5Csubseteq}{\subseteq} X [/math]
Proof: ∅ ⊆ A
This claim is vacuously true, since there are no elements of the empty set. Another more convoluted way of saying the same thing: choose an arbitrary element [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} [/math]. Then we have that [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} [/math] (by assumption) and [math]x \href{/cs2800/wiki/index.php/%5Cnotin}{\notin} \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} [/math] (by definition of [math]\emptyset [/math]). This is a contradiction, which completes this part of the proof (and thus the whole proof).