Definition df-clel 2096

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 2093 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2093 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1873), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2087.

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. (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 1448	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1298	. . . . 5 class 𝑥
6	5, 1	wceq 1299	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 1448	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 103	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1436	. 2 wff ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
10	3, 9	wb 104	1 wff (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

This definition is referenced by:  eleq1w  2160  eleq2w  2161  eleq1  2162  eleq2  2163  clelab  2224  clabel  2225  nfel  2249  nfeld  2256  sbabel  2266  risset  2422  isset  2647  elex  2652  sbcabel  2942  ssel  3041  disjsn  3532  mptpreima  4968