Definition df-clel 2439

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 2436 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2436 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2108), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2430. Alternate definitions of (but that require either or to be a set) are shown by clel2 3081, clel3 3083, and clel4 3084.

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 http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
df-clel

Distinct variable groups:   ,   ,

Detailed syntax breakdown of Definition df-clel

Step	Hyp	Ref	Expression
1		cA	. . 3
2		cB	. . 3
3	1, 2	wcel 1728	. 2
4		vx	. . . . . 6
5	4	cv 1653	. . . . 5
6	5, 1	wceq 1654	. . . 4
7	5, 2	wcel 1728	. . . 4
8	6, 7	wa 360	. . 3
9	8, 4	wex 1551	. 2
10	3, 9	wb 178	1

This definition is referenced by:  eleq1  2503  eleq2  2504  clelab  2563  clabel  2564  nfel  2587  nfeld  2594  sbabel  2605  risset  2760  isset  2969  elex  2973  sbcabel  3257  ssel  3331  disjsn  3896  pwpw0  3974  pwsnALT  4038  mptpreima  5409  brfi1uzind  11766  ballotlem2  24850  eldm3  25489