Discussion
How is there is exactly one such object expressed?
Answer: A
Uniqueness requires both existence and the statement that anything with the property is identical to the witness, which needs equality. The shorthand notation for unique existence abbreviates exactly this. The second option denies the existence of anything else at all, and the third is not well formed. Constructing a compound claim from existence plus a uniqueness clause is a pattern that recurs whenever counting is required.
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