Using the $C$% confidence interval for $\mu_1 - \mu_2$ to perform two sample $z$ test

Be sure you understand the theory behind confidence intervals and significance tests, before trying to understand the explanation behind the 'explain' button.
Two sided test If 0 is not in the $C$% confidence interval, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly different from 0 at significance level $\alpha = 1 - C/100$.
For instance, if 0 is not in the $95$% confidence interval, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly different from 0 at significance level $\alpha = 1 - 95/100 = .05$.
Explain
Right sided test If 0 is not in the $C$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is larger than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly larger than 0 at significance level $\alpha = \frac{1 \, - \, C/100}{2}$.
For instance, if 0 is not in the $90$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is larger than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly larger than 0 at significance level $\alpha = \frac{1 \,-\, 90/100}{2} = .05$.
Explain
Left sided test If 0 is not in the $C$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is smaller than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly smaller than 0 at significance level $\alpha = \frac{1 \, - \, C/100}{2}$.
For instance, if 0 is not in the $90$% confidence interval and the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is smaller than 0, the difference between the two sample means $\bar{y}_1 - \bar{y}_2$ is significantly smaller than 0 at significance level $\alpha = \frac{1 \,-\, 90/100}{2} = .05$.
Explain