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Definition df-clel 2079
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 2076 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2076 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1856), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2070.

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 1434 . 2  wff  A  e.  B
4 vx . . . . . 6  setvar  x
54cv 1284 . . . . 5  class  x
65, 1wceq 1285 . . . 4  wff  x  =  A
75, 2wcel 1434 . . . 4  wff  x  e.  B
86, 7wa 102 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1422 . 2  wff  E. x
( x  =  A  /\  x  e.  B
)
103, 9wb 103 1  wff  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Colors of variables: wff set class
This definition is referenced by:  eleq1w  2143  eleq2w  2144  eleq1  2145  eleq2  2146  clelab  2207  clabel  2208  nfel  2231  nfeld  2238  sbabel  2248  risset  2399  isset  2614  elex  2619  sbcabel  2905  ssel  3003  disjsn  3473  mptpreima  4865
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