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Definition df-clel 2113
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 2110 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2110 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1890), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2104.

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 2104. (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 1465 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1315 . . . . 5 class 𝑥
65, 1wceq 1316 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 1465 . . . 4 wff 𝑥𝐵
86, 7wa 103 . . 3 wff (𝑥 = 𝐴𝑥𝐵)
98, 4wex 1453 . 2 wff 𝑥(𝑥 = 𝐴𝑥𝐵)
103, 9wb 104 1 wff (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff set class
This definition is referenced by:  eleq1w  2178  eleq2w  2179  eleq1  2180  eleq2  2181  clelab  2242  clabel  2243  nfel  2267  nfeld  2274  sbabel  2284  risset  2440  isset  2666  elex  2671  sbcabel  2962  ssel  3061  disjsn  3555  mptpreima  5002
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