Definition df-clel 2010

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 2007 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2007 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1787), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2001.

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 1367	. 2 wff A ∈ B
4		vx	. . . . . 6 setvar x
5	4	cv 1223	. . . . 5 class x
6	5, 1	wceq 1224	. . . 4 wff x = A
7	5, 2	wcel 1367	. . . 4 wff x ∈ B
8	6, 7	wa 97	. . 3 wff (x = A ∧ x ∈ B)
9	8, 4	wex 1355	. 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  2074  eleq2  2075  clelab  2136  clabel  2137  nfel  2160  nfeld  2167  sbabel  2177  risset  2322  isset  2531  elex  2535  sbcabel  2808  ssel  2908  disjsn  3396  mptpreima  4730