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Definition df-clel 2078
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 2075 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2075 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1855), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2069.

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 (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 1434 . 2 wff 𝐴𝐵
4 vx . . . . . 6 setvar 𝑥
54cv 1284 . . . . 5 class 𝑥
65, 1wceq 1285 . . . 4 wff 𝑥 = 𝐴
75, 2wcel 1434 . . . 4 wff 𝑥𝐵
86, 7wa 102 . . 3 wff (𝑥 = 𝐴𝑥𝐵)
98, 4wex 1422 . 2 wff 𝑥(𝑥 = 𝐴𝑥𝐵)
103, 9wb 103 1 wff (𝐴𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥𝐵))
Colors of variables: wff set class
This definition is referenced by:  eleq1  2142  eleq2  2143  clelab  2204  clabel  2205  nfel  2228  nfeld  2235  sbabel  2245  risset  2395  isset  2606  elex  2611  sbcabel  2896  ssel  2994  disjsn  3462  mptpreima  4844
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