Are They Opposites?
'Anyone' refers to 'any person at all, no matter who' (we use it when it doesn't matter who the person is). Remember that In this case, the word 'anyone' is always written together as one word without any space between 'any' and 'one'.
Did you see
Is 'Any One' Correct, Too?
The only way you might see the noun phrase 'any one' (with space between them) is when we want to refer to a member of a certain group.
'Who do you think did this?' 'It could be
'No one' is another indefinite pronoun that refers to the absence of a person, i.e. it means 'not anyone; no person'. Sometimes, writers write 'no one' with a dash: 'no-one'.
'Anyone' can be used in negative sentences and questions. But, 'no one' is a negative marker by itself and it can make a sentence negative.
wasn't at home.)
The verb is positive, but the sentence is negative, because of 'no one' which is a negative marker.
The sentence is negative, because the verb is negative, not because of 'anyone'.
Can We Use 'No One of'?
Don't use 'of' after no one.
No one of us can speak Russian fluently.
Instead, you should use 'none of'.