Theorem wl-df.cleq 38351

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.

The hypotheses express that all instances of the conclusion where class variables are replaced with setvar variables hold. Therefore, this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This is only a proof sketch of conservativity; for details see Appendix of [Levy] p. 357. This is the reason why we call this axiomatic statement a "definition", even though it does not have the usual form of a definition. If we required a definition to have the usual form, we would call df-cleq 2752 an axiom.

See also comments under df-clab 2739, df-clel 2835, and eqabb 2899.

In the form of dfcleq 2753, 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. It characterizes classes as collections of sets.

While the three class definitions df-clab 2739, df-cleq 2752, and df-clel 2835 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 2835. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.)

Hypotheses

Ref	Expression
wl-df.cleq.1	⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))
wl-df.cleq.2	⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))

Assertion

Ref	Expression
wl-df.cleq	⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑡,𝑢,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑡,𝑢,𝑣

Proof of Theorem wl-df.cleq

Step	Hyp	Ref	Expression
1		wl-df.cleq.1	. 2 ⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧))
2		wl-df.cleq.2	. 2 ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))
3	1, 2	df-cleq 2752	1 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2145

This proof depends on definitions:  df-cleq 2752

This theorem is used by:  wl-dfcleq.basic  38352