Definition df-clel 2485

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 2482 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2482 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2113), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2476. Alternate definitions of (but that require either or to be a set) are shown by clel2 3134, clel3 3136, and clel4 3137.

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 1732	. 2
4		vx	. . . . . 6
5	4	cv 1669	. . . . 5
6	5, 1	wceq 1670	. . . 4
7	5, 2	wcel 1732	. . . 4
8	6, 7	wa 360	. . 3
9	8, 4	wex 1565	. 2
10	3, 9	wb 178	1

This definition is referenced by:  eleq1  2549  eleq2  2550  clelab  2609  clabel  2610  nfel  2633  nfeld  2640  sbabel  2651  risset  2807  isset  3019  elex  3023  sbcabel  3312  ssel  3387  disjsn  3968  pwpw0  4046  pwsnALT  4112  mptpreima  5351  brfi1uzind  12066  ballotlem2  26021  eldm3  26724  bj-elsngl  30950