What Does the Notation = in Equivalence Classes Conclude to?

[kidsasd987]

Please refer to the video at 37:02 from the link above.

I am struggling with the notation "=" of the property (a) which concludes to [a]=[m].
shouldn't it be [a]⊆[m], just like [m]⊆M.

 

[fresh_42]

What does $a \in [m]$ mean? And what can be said about all elements $b \sim a$ if we use transitivity of $\sim$?
In retrun, what does it mean for all elements $c \sim m$?