Definition df-clel 2228

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 2225 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2225 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2209), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2219.

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 2219. (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 2203	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1397	. . . . 5 class x
6	5, 1	wceq 1398	. . . 4 wff x = A
7	5, 2	wcel 2203	. . . 4 wff x e. B
8	6, 7	wa 104	. . 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 105	1 wff ( A e. B <-> E. x ( x = A /\ x e. B ) )

This definition is referenced by:  eleq1w  2293  eleq2w  2294  eleq1  2295  eleq2  2296  clelab  2360  clabel  2361  nfel  2393  nfeld  2400  sbabel  2411  risset  2570  isset  2819  elex  2824  sbcabel  3124  ssel  3231  disjsn  3750  mptpreima  5255