Definition df-cleq 2818

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 2807). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated.

See also comments under df-clab 2812, df-clel 2821, and abeq2 2937.

In the form of dfcleq 2819, 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 2812, df-cleq 2818, and df-clel 2821 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.)

Hypothesis

Ref	Expression
df-cleq.1	⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)

Assertion

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

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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	wceq 1656	. 2 wff 𝐴 = 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1655	. . . . 5 class 𝑥
6	5, 1	wcel 2164	. . . 4 wff 𝑥 ∈ 𝐴
7	5, 2	wcel 2164	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wb 198	. . 3 wff (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
9	8, 4	wal 1654	. 2 wff ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
10	3, 9	wb 198	1 wff (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

This definition is referenced by:  dfcleq  2819  bj-ax9  33404  bj-ax9-2  33405