You don't need the word antithetical there. The theist says there is a god and the atheist says, "I don't agree."
It doesn't matter how many times you repeat this, it's never going to become logical.
Theism proposes that God/gods exist, and this proposition then generates a given set of logical possible responses.
Affirm:
a. We accept the proposition as valid (logically and factually
viable).
b. We also accept the proposition as
true.
c. We move on to develop a theology based on this assumed truth.
Deny:
a. We do not accept the proposition.
b. We remain undecided.
c. We decide the proposition is false. (Because the proposition is either not logically or factually valid, and therefor is not true.)
The proper labels for the second group of reactions are:
a., Unspecified "non-theist".
b., "Agnostic" or perhaps "indifferent non-theist".
c., "Atheist". (Note that logically one cannot claim to be both "non-theist" and "atheist" because an atheist is not agnostic, nor indifferent, nor undecided.)