Soundness and completeness of proposition logic

Question

I managed to prove that

1. a set $\Gamma $ is inconsistent if and only if $\Gamma \vdash a$ where $a$ is an arbitrary wff
2. a set $\Gamma $ is inconsistent if and only if $\Gamma \vDash b$ where $b$ is an arbitrary wff

Is it appropriate make a conclusion that the selected axiomatization of propositional logic is strongly sound and strongly complete (e.g., $\Gamma \vdash c$ if and only if $\Gamma \vDash c$) based solely on the two statements above?