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Let $A$ and $B$ be classes.
Let $B$ be a subclass of $A$.
Then $B$ is a proper subclass of $A$ if and only if $B \ne A$.
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.2$