Summary: Motivated by Leibniz's thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets \(x\) and~\(y\) there exists an ordinal \(\alpha\) exceeding the ranks of \(x\) and~\(y\), and a formula \(\varphi(v),\) such that \((V_{\alpha},\in)\) satisfies \(\varphi(x)\wedge\lnot\varphi(y)\). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: (1) In the presence of ZF, the following are equivalent: (a) LM. (b) The existence of a parameter-free definable class function \textbf{F} such that for all sets \(x\) with at least two elements, \(\emptyset \neq {\mathbf F}(x)\subsetneq x.\) (c) The existence of a parameter-free definable injection of the universe into the class of subsets of ordinals. (2) \(\text{Con(ZF)} \Rightarrow \text{Con(ZFC}+\lnot \text{LM})\). (3) [Solovay] \(\text{Con(ZF)} \Rightarrow \text{Con(ZF}+ \text{LM} +\lnot \text{AC})\).