Definition df-clel 2135

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 2132 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2132 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1910), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2126.

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 2126. (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 1480	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1330	. . . . 5 class x
6	5, 1	wceq 1331	. . . 4 wff x = A
7	5, 2	wcel 1480	. . . 4 wff x e. B
8	6, 7	wa 103	. . 3 wff ( x = A /\ x e. B )
9	8, 4	wex 1468	. 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  2200  eleq2w  2201  eleq1  2202  eleq2  2203  clelab  2265  clabel  2266  nfel  2290  nfeld  2297  sbabel  2307  risset  2463  isset  2692  elex  2697  sbcabel  2990  ssel  3091  disjsn  3585  mptpreima  5032