Definition 37: For classes x and y , we shall say x equals y if for all sets z we have that z \in x if and only if z \in y . In place of “equals”, we may say “is equal to“, “is the same as” or just “is”. We introduce the binary predicate (x \hspace{1mm} \underset{w}{\stackrel{z}{=}} \hspace{1mm} y) \hspace{3mm} \textup{defined by} \hspace{3mm} \forall z \Big( \underset{w}{\textup{Set}}(z) \Rightarrow \big( (z \in x) \Leftrightarrow (z \in y) \big) \Big), where the right side is a bounded universal statement that involves the \textup{Set} predicate, equivalence and membership statements, and means x equals y . Should no confusion arise, we shall generally write (x = y) \hspace{3mm} \textup{in place of} \hspace{3mm} (x \hspace{1mm} \underset{w}{\stackrel{z}{=}} \hspace{1mm} y), taking for granted that w and z are distinct from all other variables involved. We call such a statement as an equality or equation. The negation of (x = y) is written using the standard notation for negation of binary predicates as (x \neq y), and may be expressed by saying x and y are distinct or not equal.