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A set $A$ is well-ordered with respect to the membership relation if the membership relation $\in_A$ over $A$,

$$\in_A=\{\langle a,b\rangle\in A\times A\vert a\in b\}$$

is a (strict) well-order in $A$, that is, for any $a$, $b$, $c\in A$,

  1. $a\not\in a$
  2. if $a\in b$ and $b\in c$, then $a\in c$
  3. $a\in b$ or $b\in a$ or $a=b$
  4. each non-empty subset $B$ in $A$ has an element $d$ such that $d\in b$ for any $b\in B$ different from $d$

A set $A$ is an ordinal (number) if it is transitive and well-ordered with respect to the relation of set membership $\in_A$