Definition df-clel 2036

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 2033 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2033 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1813), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2027.

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 http://us.metamath.org/mpeuni/mmset.html#class. (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 1393	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1242	. . . . 5 class 𝑥
6	5, 1	wceq 1243	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 1393	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 97	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1381	. 2 wff ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
10	3, 9	wb 98	1 wff (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

This definition is referenced by:  eleq1  2100  eleq2  2101  clelab  2162  clabel  2163  nfel  2186  nfeld  2193  sbabel  2203  risset  2352  isset  2561  elex  2566  sbcabel  2839  ssel  2939  disjsn  3432  mptpreima  4814