Definition df-clel 2019

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 2016 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2016 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 2010.

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 1376	. 2 wff A ∈ B
4		vx	. . . . . 6 setvar x
5	4	cv 1373	. . . . 5 class x
6	5, 1	wceq 1374	. . . 4 wff x = A
7	5, 2	wcel 1376	. . . 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  2083  eleq2  2084  clelab  2145  clabel  2146  nfel  2169  nfeld  2176  sbabel  2186  risset  2329  isset  2538  elex  2542  sbcabel  2816  ssel  2918  disjsn  3384  mptpreima  4718