Definition df-clel 2136

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 2133 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2133 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1911), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2127.

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 2127. (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 1481	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1331	. . . . 5 class 𝑥
6	5, 1	wceq 1332	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 1481	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 103	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1469	. 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  2201  eleq2w  2202  eleq1  2203  eleq2  2204  clelab  2266  clabel  2267  nfel  2291  nfeld  2298  sbabel  2308  risset  2466  isset  2695  elex  2700  sbcabel  2994  ssel  3096  disjsn  3593  mptpreima  5040