Definition df-clel 2166

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 2163 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2163 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2147), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2157.

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 2157. (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 2141	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1347	. . . . 5 class 𝑥
6	5, 1	wceq 1348	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 2141	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 103	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1485	. 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  2231  eleq2w  2232  eleq1  2233  eleq2  2234  clelab  2296  clabel  2297  nfel  2321  nfeld  2328  sbabel  2339  risset  2498  isset  2736  elex  2741  sbcabel  3036  ssel  3141  disjsn  3645  mptpreima  5104