Question

Is it acceptable to say that something is equal to something else "by definition" without saying what the definition is?

Answer 1

depends on how much common knowledgic the definition is :)

Answer 2

The problem:
Prove: If A is a subset of the null set ∅, then A = ∅.

The solution:
The null set ∅ is a subset of every set; in particular, ∅ ⊂ A. But, by hypothesis, A ⊂ ∅; hence, by Definition 1.1, A = ∅.

The definition:
Definition 1.1: Two sets A and B are equal if and only if A ⊂ B and B ⊂ A.

Answer 3

In this case, would it be acceptable to just say "by definition"?