Definition df-clel 2161

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 2158 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2158 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2142), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2152.

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 2152. (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 2136	. 2 wff A e. B
4		vx	. . . . . 6 setvar x
5	4	cv 1342	. . . . 5 class x
6	5, 1	wceq 1343	. . . 4 wff x = A
7	5, 2	wcel 2136	. . . 4 wff x e. B
8	6, 7	wa 103	. . 3 wff ( x = A /\ x e. B )
9	8, 4	wex 1480	. 2 wff E. x ( x = A /\ x e. B )
10	3, 9	wb 104	1 wff ( A e. B <-> E. x ( x = A /\ x e. B ) )

This definition is referenced by:  eleq1w  2227  eleq2w  2228  eleq1  2229  eleq2  2230  clelab  2292  clabel  2293  nfel  2317  nfeld  2324  sbabel  2335  risset  2494  isset  2732  elex  2737  sbcabel  3032  ssel  3136  disjsn  3638  mptpreima  5097