$\leq_R$ is a well­order on $A$:
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$ \forall B($$\scriptsize \neq \phi)$$\subset A \exists b_0\in B(b_0$ is a minimum element$)$

Well­order


$A,\leq_R$ is a Well­ordered set: $\leq_R$ is wellorder on $A$

Well­ordered set


$\leq_{N}$ is lexicographical order of $\mathbb{N} \times \mathbb{N}$:
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$\small (n_1,n_2)\leq_{N} (n_1',n_2')\Leftrightarrow n_1< _Nn_1' \lor (n_1=n_1' \land n_2\leq_N n_2')$

Lexicographical order

$\forall a\in A (\forall b\in A(b<a \rightarrow P(b)) \rightarrow P(a)))$
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$\Rightarrow \forall a\in A(p(a))$

Wellordered set

$Wellorder$

$Wellordered set$

$Lexicographical order$

$Transfinite induction$