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Definition df-clel 2014
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 2011 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2011 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1791), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2005.

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
Distinct variable groups:   ,   ,

Detailed syntax breakdown of Definition df-clel
StepHypRef Expression
1 cA . . 3
2 cB . . 3
31, 2wcel 1370 . 2
4 vx . . . . . 6  setvar
54cv 1225 . . . . 5
65, 1wceq 1226 . . . 4
75, 2wcel 1370 . . . 4
86, 7wa 97 . . 3
98, 4wex 1358 . 2
103, 9wb 98 1
Colors of variables: wff set class
This definition is referenced by:  eleq1  2078  eleq2  2079  clelab  2140  clabel  2141  nfel  2164  nfeld  2171  sbabel  2181  risset  2326  isset  2535  elex  2539  sbcabel  2812  ssel  2912  disjsn  3402  mptpreima  4737
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