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Definition df-cleq 2075
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(𝑥𝑦𝑥𝑧), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2066). 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 =2, 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 =2 𝑦𝑥 = 𝑦 by invoking Extensionality.

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 2069, df-clel 2078, and abeq2 2188.

In the form of dfcleq 2076, 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.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 15-Sep-1993.)

Hypothesis
Ref Expression
df-cleq.1 (∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)
Assertion
Ref Expression
df-cleq (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐵(𝑦,𝑧)

Detailed syntax breakdown of Definition df-cleq
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2wceq 1285 . 2 wff 𝐴 = 𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1284 . . . . 5 class 𝑥
65, 1wcel 1434 . . . 4 wff 𝑥𝐴
75, 2wcel 1434 . . . 4 wff 𝑥𝐵
86, 7wb 103 . . 3 wff (𝑥𝐴𝑥𝐵)
98, 4wal 1283 . 2 wff 𝑥(𝑥𝐴𝑥𝐵)
103, 9wb 103 1 wff (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
This definition is referenced by:  dfcleq  2076
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