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Definition df-clel 2185
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 2182 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2182 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2166), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2176.

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 2176. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
df-clel  |-  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  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 2160 . 2  wff  A  e.  B
4 vx . . . . . 6  setvar  x
54cv 1363 . . . . 5  class  x
65, 1wceq 1364 . . . 4  wff  x  =  A
75, 2wcel 2160 . . . 4  wff  x  e.  B
86, 7wa 104 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1503 . 2  wff  E. x
( x  =  A  /\  x  e.  B
)
103, 9wb 105 1  wff  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Colors of variables: wff set class
This definition is referenced by:  eleq1w  2250  eleq2w  2251  eleq1  2252  eleq2  2253  clelab  2315  clabel  2316  nfel  2341  nfeld  2348  sbabel  2359  risset  2518  isset  2758  elex  2763  sbcabel  3059  ssel  3164  disjsn  3669  mptpreima  5140
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