Definition df-clel 2160

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 2157 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2157 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2141), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2151.

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 2151. (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 2135	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1341	. . . . 5 class x
6	5, 1	wceq 1342	. . . 4 wff x = A
7	5, 2	wcel 2135	. . . 4 wff x e. B
8	6, 7	wa 103	. . 3 wff ( x = A /\ x e. B )
9	8, 4	wex 1479	. 2 wff E. x ( x = A /\ x e. B )
10	3, 9	wb 104	1 wff ( A e. B <-> E. x ( x = A /\ x e. B ) )

This definition is referenced by:  eleq1w  2225  eleq2w  2226  eleq1  2227  eleq2  2228  clelab  2290  clabel  2291  nfel  2315  nfeld  2322  sbabel  2333  risset  2492  isset  2727  elex  2732  sbcabel  3027  ssel  3131  disjsn  3632  mptpreima  5091