Definition df-clel 2234

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 2231 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2231 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2215), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2225.

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 2225. (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 2209	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1401	. . . . 5 class 𝑥
6	5, 1	wceq 1402	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 2209	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 104	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1545	. 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  2299  eleq2w  2300  eleq1  2301  eleq2  2302  clelab  2366  clabel  2367  nfel  2401  nfeld  2408  sbabel  2419  risset  2578  isset  2828  elex  2833  sbcabel  3134  ssel  3242  disjsn  3770  mptpreima  5279