Definition df-clel 2081

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 2078 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2078 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1858), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2072.

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 e. B <-> E. x ( x = A /\ x e. 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 1436	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1286	. . . . 5 class x
6	5, 1	wceq 1287	. . . 4 wff x = A
7	5, 2	wcel 1436	. . . 4 wff x e. B
8	6, 7	wa 102	. . 3 wff ( x = A /\ x e. B )
9	8, 4	wex 1424	. 2 wff E. x ( x = A /\ x e. B )
10	3, 9	wb 103	1 wff ( A e. B <-> E. x ( x = A /\ x e. B ) )

This definition is referenced by:  eleq1w  2145  eleq2w  2146  eleq1  2147  eleq2  2148  clelab  2209  clabel  2210  nfel  2233  nfeld  2240  sbabel  2250  risset  2402  isset  2619  elex  2624  sbcabel  2909  ssel  3008  disjsn  3487  mptpreima  4890