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Definition df-clel 2173
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 2170 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2170 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2154), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2164.

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 2164. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
df-clel (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Detailed syntax breakdown of Definition df-clel
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2wcel 2148 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1352 . . . . 5 class 𝑥
65, 1wceq 1353 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 2148 . . . 4 wff 𝑥𝐵
86, 7wa 104 . . 3 wff (𝑥 = 𝐴𝑥𝐵)
98, 4wex 1492 . 2 wff 𝑥(𝑥 = 𝐴𝑥𝐵)
103, 9wb 105 1 wff (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff set class
This definition is referenced by:  eleq1w  2238  eleq2w  2239  eleq1  2240  eleq2  2241  clelab  2303  clabel  2304  nfel  2328  nfeld  2335  sbabel  2346  risset  2505  isset  2743  elex  2748  sbcabel  3044  ssel  3149  disjsn  3654  mptpreima  5122
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