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
rather
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 2604). We therefore include this axiom as a
hypothesis, so that
the use of Extensionality is properly indicated.
See also comments under df-clab 2607, df-clel 2616, and abeq2 2730.
In the form of dfcleq 2614, this is called the "axiom of
extensionality"
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 2607, df-cleq 2613, and df-clel 2616
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,
15-Sep-1993.) |