$\displaystyle [\![ \alpha \subset z ]\!]= \inf_{\beta \in \text{dom}(\alpha)} (\alpha(\beta)\Longrightarrow [\![ \beta \in z ]\!]) = \inf_{\beta \in \text{dom}(\alpha)} ([\![\beta \in z]\!]\Longrightarrow [\![\beta \in z]\!])=1$.

Por outro lado, dado $\beta$ nome qualquer

$[\![\beta \in \dot{x} \wedge \beta \in z]\!]$

$\displaystyle = \sup_{t \in \text{dom}(\dot{x})}\dot{x}(t)[\![t=\beta]\!][\![\beta \in z]\!]$

$\displaystyle =\sup_{t \in \text{dom}(\dot{x})}\dot{x}(t)[\![t=\beta]\!][\![\beta \in z]\!][\![t=\beta]\!]$

$\displaystyle \leq \sup_{t \in \text{dom}(\dot{x})}\dot{x}(t)[\![t \in z]\!][\![t=\beta]\!]$

$\displaystyle \leq \sup_{t \in \text{dom}(\alpha)}[\![t \in z]\!][\![t=\beta]\!]$

$\displaystyle \leq \sup_{t \in \text{dom}(\alpha)}\alpha(t)[\![t=\beta]\!]$

$=[\![\beta \in \alpha]\!]$

Logo $[\![(\dot{x}\cap z)\subset \alpha]\!]=1$.

Então $[\![z \subset \dot{x}]\!]=[\![z \subset \dot{x}]\!][\![(\dot{x}\cap z)\subset \alpha]\!]\leq[\![z \subset \alpha]\!]$.

Portanto, $[\![z\subset \dot{x} \Longrightarrow z = \alpha]\!]=1$.