Definition df-clel 2171

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 2168 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2168 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2152), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2162.

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 2162. (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 2146	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1352	. . . . 5 class x
6	5, 1	wceq 1353	. . . 4 wff x = A
7	5, 2	wcel 2146	. . . 4 wff x e. B
8	6, 7	wa 104	. . 3 wff ( x = A /\ x e. B )
9	8, 4	wex 1490	. 2 wff E. x ( x = A /\ x e. B )
10	3, 9	wb 105	1 wff ( A e. B <-> E. x ( x = A /\ x e. B ) )

This definition is referenced by:  eleq1w  2236  eleq2w  2237  eleq1  2238  eleq2  2239  clelab  2301  clabel  2302  nfel  2326  nfeld  2333  sbabel  2344  risset  2503  isset  2741  elex  2746  sbcabel  3042  ssel  3147  disjsn  3651  mptpreima  5114