At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)

Any IVP admits an upper and a lower solution, whose domains are maximal intervals. For $(t_0,x_0)\in\Omega$, define, for $t\in\mathbb{R}$ $$x ^ * (t):=\sup\{x(t)\\ : x\in C^1(\mathrm{co}(t_0,t),\\ \mathbb{R}),\\ \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s)) \}\\ ,$$ (where of course $\sup\emptyset=-\infty$). Define $\mathrm{dom}(x ^ *)$ to be the connected component of $t_0$ in the set $\{ t: x ^ *(t) \in\mathbb{R} \}$. Then, $x ^ *$ is a solution of the ODE with IVC $x(t_0)=x_0$, maximally defined on the interval $\mathrm{dom}(x ^ *)$.

At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)

Any IVP admits an upper and a lower solution, whose domains are maximal intervals. For $(t_0,x_0)\in\Omega$, define, for $t\in\mathbb{R}$ $$x ^ * (t):=\sup\big\{x(t)\, : x\in C^1(\mathrm{co}(t_0,t),\, \mathbb{R}),\, \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s)) \big\}\, ,$$ (where of course $\sup\emptyset=-\infty$). Define $\mathrm{dom}(x ^ *)$ to be the connected component of $t_0$ in the set $\{ t: x ^ *(t) \in\mathbb{R} \}$. Then, $x ^ *$ is a solution of the ODE with IVC $x(t_0)=x_0$, maximally defined on the interval $\mathrm{dom}(x ^ *)$.

At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)

Any IVP admits an upper and a lower solution, whose domains are maximal intervals. For $(t_0,x_0)\in\Omega$, define, for $t\in\mathbb{R}$ $$x ^ * (t):=\sup\{x(t)\\ : x\in C^1(\mathrm{co}(t_0,t),\\ \mathbb{R}),\\ \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s)) \}\\ ,$$ (where of course $\sup\emptyset=-\infty$). Define $\mathrm{dom}(x ^ *)$ to be the connected component of $t_0$ in the set $\{ t: x ^ *(t) > -\infty\}$. Then, $x ^ *$ is a solution of the ODE with IVC $x(t_0)=x_0$, maximally defined on the interval $\mathrm{dom}(x ^ *)$.

At least for scalar equations $\dot x(t)=f(t,x(t))$, that is with a nonlinearity $f\in C^0(\Omega,\mathbb{R})$, defined on an open set $\Omega\subset \mathbb{R}^2$, the Zorn's lemma is not necessary: the order structure of $\mathbb{R}$ allows to select a preferred solution (actually, two)

Any IVP admits an upper and a lower solution, whose domains are maximal intervals. For $(t_0,x_0)\in\Omega$, define, for $t\in\mathbb{R}$ $$x ^ * (t):=\sup\{x(t)\\ : x\in C^1(\mathrm{co}(t_0,t),\\ \mathbb{R}),\\ \mathrm{graph}(x)\subset\Omega, x(t_0)=x_0, \dot x(s)=f(s,x(s)) \}\\ ,$$ (where of course $\sup\emptyset=-\infty$). Define $\mathrm{dom}(x ^ *)$ to be the connected component of $t_0$ in the set $\{ t: x ^ *(t) \in\mathbb{R} \}$. Then, $x ^ *$ is a solution of the ODE with IVC $x(t_0)=x_0$, maximally defined on the interval $\mathrm{dom}(x ^ *)$.