Definition df-clel 2077

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 2074 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2074 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1854), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2068.

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 1433	. 2 wff 𝐴 ∈ 𝐵
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1283	. . . . 5 class 𝑥
6	5, 1	wceq 1284	. . . 4 wff 𝑥 = 𝐴
7	5, 2	wcel 1433	. . . 4 wff 𝑥 ∈ 𝐵
8	6, 7	wa 102	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1421	. 2 wff ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
10	3, 9	wb 103	1 wff (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

This definition is referenced by:  eleq1  2141  eleq2  2142  clelab  2203  clabel  2204  nfel  2227  nfeld  2234  sbabel  2244  risset  2394  isset  2605  elex  2610  sbcabel  2895  ssel  2993  disjsn  3454  mptpreima  4834