|Description: Define the equality
connective between classes. Definition 2.7 of
[Quine] p. 18. Also Definition 4.5 of
[TakeutiZaring] p. 13; Chapter 4
provides its justification and methods for eliminating it. Note that
its elimination will not necessarily result in a single wff in the
original language but possibly a "scheme" of wffs.
This is an example of a somewhat "risky" definition, meaning
that it has
a more complex than usual soundness justification (outside of Metamath),
because it "overloads" or reuses the existing equality symbol
than introducing a new symbol. This allows us to make statements that
may not hold for the original symbol. For example, it permits us to
deduce 𝑦 = 𝑧 ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧), which is not a theorem
of logic but rather presupposes the Axiom of Extensionality (see theorem
axext4 2489). We therefore include this axiom as a
hypothesis, so that
the use of Extensionality is properly indicated.
We could avoid this complication by introducing a new symbol, say
in place of =. This would also have the
advantage of making
elimination of the definition straightforward, so that we could
eliminate Extensionality as a hypothesis. We would then also have the
advantage of being able to identify in various proofs exactly where
Extensionality truly comes into play rather than just being an artifact
of a definition. One of our theorems would then be 𝑥
↔ 𝑥 = 𝑦 by invoking
However, to conform to literature usage, we retain this overloaded
definition. This also makes some proofs shorter and probably easier to
read, without the constant switching between two kinds of equality.
See also comments under df-clab 2492, df-clel 2501, and abeq2 2614.
In the form of dfcleq 2499, this is called the "axiom of
by [Levy] p. 338, who treats the theory of
classes as an extralogical
extension to our logic and set theory axioms.
While the three class definitions df-clab 2492, df-cleq 2498, and df-clel 2501
are eliminable and conservative and thus meet the requirements for sound
definitions, they are technically axioms in that they do not satisfy the
requirements for the current definition checker. The proofs of
conservativity require external justification that is beyond the scope
of the definition checker.
For a general discussion of the theory of classes, see
mmset.html#class. (Contributed by NM,