Definition df-clel 2192

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. Note that like df-cleq 2189 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2189 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2173), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2183.

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.

For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2183. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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 2167	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1363	. . . . 5 class 𝑥
6	5, 1	wceq 1364	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 2167	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 104	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1506	. 2 wff ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
10	3, 9	wb 105	1 wff (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

This definition is referenced by:  eleq1w  2257  eleq2w  2258  eleq1  2259  eleq2  2260  clelab  2322  clabel  2323  nfel  2348  nfeld  2355  sbabel  2366  risset  2525  isset  2769  elex  2774  sbcabel  3071  ssel  3177  disjsn  3684  mptpreima  5163