Definition df-clel 2018

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 2015 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2015 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1795), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2009.

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 ∈ B ↔ ∃x(x = A ∧ x ∈ 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 1374	. 2 wff A ∈ B
4		vx	. . . . . 6 setvar x
5	4	cv 1227	. . . . 5 class x
6	5, 1	wceq 1228	. . . 4 wff x = A
7	5, 2	wcel 1374	. . . 4 wff x ∈ B
8	6, 7	wa 97	. . 3 wff (x = A ∧ x ∈ B)
9	8, 4	wex 1362	. 2 wff ∃x(x = A ∧ x ∈ B)
10	3, 9	wb 98	1 wff (A ∈ B ↔ ∃x(x = A ∧ x ∈ B))

This definition is referenced by:  eleq1  2082  eleq2  2083  clelab  2144  clabel  2145  nfel  2168  nfeld  2175  sbabel  2185  risset  2330  isset  2539  elex  2543  sbcabel  2816  ssel  2916  disjsn  3406  mptpreima  4741