Definition df-clel 2202

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 2199 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2199 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2183), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2193.

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 2193. (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 2177	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1372	. . . . 5 class 𝑥
6	5, 1	wceq 1373	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 2177	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 104	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1516	. 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  2267  eleq2w  2268  eleq1  2269  eleq2  2270  clelab  2332  clabel  2333  nfel  2358  nfeld  2365  sbabel  2376  risset  2535  isset  2780  elex  2785  sbcabel  3082  ssel  3189  disjsn  3697  mptpreima  5182