Definition df-clel 2033

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 2030 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2030 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1810), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2024.

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 1390	. 2 wff A ∈ B
4		vx	. . . . . 6 setvar x
5	4	cv 1241	. . . . 5 class x
6	5, 1	wceq 1242	. . . 4 wff x = A
7	5, 2	wcel 1390	. . . 4 wff x ∈ B
8	6, 7	wa 97	. . 3 wff (x = A ∧ x ∈ B)
9	8, 4	wex 1378	. 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  2097  eleq2  2098  clelab  2159  clabel  2160  nfel  2183  nfeld  2190  sbabel  2200  risset  2346  isset  2555  elex  2560  sbcabel  2833  ssel  2933  disjsn  3423  mptpreima  4757