Definition df-clel 2225

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 2222 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2222 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2206), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2216.

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 2216. (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 2200	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1394	. . . . 5 class 𝑥
6	5, 1	wceq 1395	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 2200	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 104	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1538	. 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  2290  eleq2w  2291  eleq1  2292  eleq2  2293  clelab  2355  clabel  2356  nfel  2381  nfeld  2388  sbabel  2399  risset  2558  isset  2806  elex  2811  sbcabel  3111  ssel  3218  disjsn  3728  mptpreima  5218