horizontal line thru conf int

[ekbrown77]

I have the idea that if I put a regression line through data points in a scatterplot, with a 95% confidence interval drawn around the regression line, and I can draw a horizontal line completely within the 95% CI band, that the regression line is likely not significantly sloped. Contrarily, if the horizontal line extends outside of the CI band on the left and right ends of the CI band, then the regression line is likely significantly sloped.

Is this the case? I thought I read this in SFLWR, but I can't find it now. Thanks.