Axiom of Foundation (Strong Form)/Proof 1
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
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.20$