I would like to prove De Morgan’s laws and have tried to follow the Wikipedia proof. However, I am stuck in part 1 of this proof, line 3:
1: Let $x \in (A \cap B)^c $. Then, $ x \notin A \cap B $.
2: Because $ A \cap B = \{y | y \in A \wedge y \in B \}$, it must be the case that $ x \notin A$ or $x \notin B$.
3: If $x \notin A$, then $x \in A^c$, so $x \in A^c \cup B^c$
Why is the part after the comma (“so, ...”) correct and where is it coming from?
Thank you.
If $x$ belongs to a set, then it belongs to every larger set. So, if $x\in A^\complement$, then it also belongs to $A^\complement\cup B^\complement$.