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Definition df-clel 2010
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 2007 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2007 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1787), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2001.

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 Bx(x = A x B))
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 1367 . 2 wff A B
4 vx . . . . . 6 setvar x
54cv 1223 . . . . 5 class x
65, 1wceq 1224 . . . 4 wff x = A
75, 2wcel 1367 . . . 4 wff x B
86, 7wa 97 . . 3 wff (x = A x B)
98, 4wex 1355 . 2 wff x(x = A x B)
103, 9wb 98 1 wff (A Bx(x = A x B))
Colors of variables: wff set class
This definition is referenced by:  eleq1  2074  eleq2  2075  clelab  2136  clabel  2137  nfel  2160  nfeld  2167  sbabel  2177  risset  2322  isset  2531  elex  2535  sbcabel  2808  ssel  2908  disjsn  3396  mptpreima  4730
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