Definition df-clel 2349

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 2346 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2346 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2014), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2340. Alternate definitions of A e. B (but that require either A or B to be a set) are shown by clel2 2975, clel3 2977, and clel4 2978.

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	|- (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 1710	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1641	. . . . 5 class x
6	5, 1	wceq 1642	. . . 4 wff x = A
7	5, 2	wcel 1710	. . . 4 wff x e. B
8	6, 7	wa 358	. . 3 wff (x = A /\ x e. B)
9	8, 4	wex 1541	. 2 wff E.x(x = A /\ x e. B)
10	3, 9	wb 176	1 wff (A e. B <-> E.x(x = A /\ x e. B))

This definition is referenced by:  eleq1  2413  eleq2  2414  clelab  2473  clabel  2474  nfel  2497  nfeld  2504  sbabel  2515  risset  2661  isset  2863  elex  2867  sbcabel  3123  ssel  3267  disjsn  3786  pwpw0  3855  pwsnALT  3882  axcnvprim  4091  axssetprim  4092  axsiprim  4093  axtyplowerprim  4094  axins2prim  4095  axins3prim  4096  dfnnc2  4395  nnsucelrlem1  4424  ncfinraiselem2  4480  ncfinlowerlem1  4482  eqtfinrelk  4486  nnadjoinlem1  4519  nnpweqlem1  4522  srelk  4524  tfinnnlem1  4533  opelxp  4811  mptpreima  5682  composeex  5820  domfnex  5870  ranfnex  5871  transex  5910  refex  5911  antisymex  5912  connexex  5913  foundex  5914  extex  5915  symex  5916  ceexlem1  6173  tcfnex  6244  nchoicelem11  6299  nchoicelem16  6304