Definition df-clel 2810

Description: Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification.

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-clel 2810 an axiom.

See also comments under df-clab 2714, df-cleq 2727, and eqabb 2874.

Alternate characterizations of 𝐴 ∈ 𝐵 when either 𝐴 or 𝐵 is a set are given by clel2g 3599, clel3g 3601, and clel4g 3603.

This is called the "axiom of membership" 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 2714, df-cleq 2727, and df-clel 2810 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 2810. (Contributed by NM, 26-May-1993.) (Revised by BJ, 27-Jun-2019.)

Hypotheses

Ref	Expression
df-clel.1	⊢ (𝑦 ∈ 𝑧 ↔ ∃𝑢(𝑢 = 𝑦 ∧ 𝑢 ∈ 𝑧))
df-clel.2	⊢ (𝑡 ∈ 𝑡 ↔ ∃𝑣(𝑣 = 𝑡 ∧ 𝑣 ∈ 𝑡))

Assertion

Ref	Expression
df-clel	⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

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

Detailed syntax breakdown of Definition df-clel

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	wcel 2114	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1541	. . . . 5 class 𝑥
6	5, 1	wceq 1542	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 2114	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 395	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1781	. 2 wff ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
10	3, 9	wb 206	1 wff (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

This definition is referenced by:  dfclel  2811  wl-df.clel  37815