Definition df-clel 2203

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 2200 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2200 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2184), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2194.

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 2194. (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 2178	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1372	. . . . 5 class x
6	5, 1	wceq 1373	. . . 4 wff x = A
7	5, 2	wcel 2178	. . . 4 wff x e. B
8	6, 7	wa 104	. . 3 wff ( x = A /\ x e. B )
9	8, 4	wex 1516	. 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  2268  eleq2w  2269  eleq1  2270  eleq2  2271  clelab  2333  clabel  2334  nfel  2359  nfeld  2366  sbabel  2377  risset  2536  isset  2783  elex  2788  sbcabel  3088  ssel  3195  disjsn  3705  mptpreima  5195