I need to prove $A\cup B = \Omega \Longleftrightarrow{A^c \subseteq B}$
How can I do it? This is what I've got so far, but I don't know if it is a valid demonstration:
$A\cup B = \Omega \Longleftrightarrow{x\in{A}\vee x\in{B}}\Longleftrightarrow{x\notin{A}\rightarrow{x\in{B}}}\Longleftrightarrow{A^c\subseteq B}$
Is this correct? If not, how can I prove that? Thanks in advanced.