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Definition df-clel 2400
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 2397 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2397 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2064), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2391. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 3032, clel3 3034, and clel4 3035.

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 1721 . 2  wff  A  e.  B
4 vx . . . . . 6  set  x
54cv 1648 . . . . 5  class  x
65, 1wceq 1649 . . . 4  wff  x  =  A
75, 2wcel 1721 . . . 4  wff  x  e.  B
86, 7wa 359 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1547 . 2  wff  E. x
( x  =  A  /\  x  e.  B
)
103, 9wb 177 1  wff  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Colors of variables: wff set class
This definition is referenced by:  eleq1  2464  eleq2  2465  clelab  2524  clabel  2525  nfel  2548  nfeld  2555  sbabel  2566  risset  2713  isset  2920  elex  2924  sbcabel  3198  ssel  3302  disjsn  3828  pwpw0  3906  pwsnALT  3970  mptpreima  5322  brfi1uzind  11670  ballotlem2  24699  eldm3  25333
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