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Definition df-clel 2052
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 2049 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2049 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1829), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2043.

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  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 1409 . 2  wff  A  e.  B
4 vx . . . . . 6  setvar  x
54cv 1258 . . . . 5  class  x
65, 1wceq 1259 . . . 4  wff  x  =  A
75, 2wcel 1409 . . . 4  wff  x  e.  B
86, 7wa 101 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1397 . 2  wff  E. x
( x  =  A  /\  x  e.  B
)
103, 9wb 102 1  wff  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Colors of variables: wff set class
This definition is referenced by:  eleq1  2116  eleq2  2117  clelab  2178  clabel  2179  nfel  2202  nfeld  2209  sbabel  2219  risset  2369  isset  2578  elex  2583  sbcabel  2866  ssel  2966  disjsn  3459  mptpreima  4841
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