Definition df-clel 2230

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 2227 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2227 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2211), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2221.

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 2221. (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 2205	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1397	. . . . 5 class 𝑥
6	5, 1	wceq 1398	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 2205	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 104	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1541	. 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  2295  eleq2w  2296  eleq1  2297  eleq2  2298  clelab  2362  clabel  2363  nfel  2395  nfeld  2402  sbabel  2413  risset  2572  isset  2822  elex  2827  sbcabel  3127  ssel  3234  disjsn  3753  mptpreima  5258