
Direct product of $\{A_\lambda\}_{\lambda \in \Lambda}$\\$\displaystyle{\prod_{\lambda \in \Lambda}}A_\lambda:=\{f:\Lambda \rightarrow \displaystyle{\bigcup_{\lambda \in \Lambda}A_\lambda}|\forall \lambda \in \Lambda \ (f(\lambda )\in A_\lambda)\}$

Direct product of a family of set

$ \Lambda\neq \phi \land \forall \lambda \in \Lambda (A_\lambda \neq \phi)\Rightarrow\displaystyle{\prod_{\lambda \in \Lambda}}A_\lambda \neq \phi$

 axiom of choice


$(A,\leq_R)$ is an inductively ordered set:\\$B\subset A, B\neq \phi, (B,\leq_{R})$ total ordered set$\Rightarrow B$ has upper bound in $A$

Inductively ordered set

All inductive ordered sets have a maximal element


Zorn's lemma

Well-ordering theorem

Axiom of choice $\Leftrightarrow$ Zorn's Lemma $\Leftrightarrow$ well-ordering theorem

Axiom of Choice

$Direct product of a family of set$

$axiom of choice$

$Inductively ordered set$

$Zorn's lemma$

$Well-ordering theorem$

$Proposition$