Definition df-clel 2201

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 2198 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2198 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2182), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2192.

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 2192. (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 2176	. 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 2176	. . . 4 wff x e. B
8	6, 7	wa 104	. . 3 wff ( x = A /\ x e. B )
9	8, 4	wex 1515	. 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  2266  eleq2w  2267  eleq1  2268  eleq2  2269  clelab  2331  clabel  2332  nfel  2357  nfeld  2364  sbabel  2375  risset  2534  isset  2778  elex  2783  sbcabel  3080  ssel  3187  disjsn  3695  mptpreima  5176