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 ∈ 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 2136	. 2 wff 𝐴 ∈ 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 ∈ 𝐵
8	6, 7	wa 103	. . 3 wff (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
9	8, 4	wex 1480	. 2 wff ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)
10	3, 9	wb 104	1 wff (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

This definition is referenced by:  eleq1w  2226  eleq2w  2227  eleq1  2228  eleq2  2229  clelab  2291  clabel  2292  nfel  2316  nfeld  2323  sbabel  2334  risset  2493  isset  2731  elex  2736  sbcabel  3031  ssel  3135  disjsn  3637  mptpreima  5096