Axiom of Foundation (Strong Form)/Proof 1

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Theorem

Let $B$ be a class.

Suppose $B$ is non-empty.


Then $B$ has a strictly minimal element under $\in$.


Proof

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By Epsilon Relation is Strictly Well-Founded, $\Epsilon$, the epsilon relation, is a strictly well-founded relation on $B$.

By Epsilon Relation is Proper, $\struct {\mathbb U, \Epsilon}$ is a proper relational structure, where $\mathbb U$ is the universal class.

By Well-Founded Proper Relational Structure Determines Minimal Elements, $B$ has a strictly minimal element under $\Epsilon$.

$\blacksquare$


Sources