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Definition df-clel 2432
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 2429 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2429 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2103), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2423. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 3072, clel3 3074, and clel4 3075.

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 1725 . 2  wff  A  e.  B
4 vx . . . . . 6  set  x
54cv 1651 . . . . 5  class  x
65, 1wceq 1652 . . . 4  wff  x  =  A
75, 2wcel 1725 . . . 4  wff  x  e.  B
86, 7wa 359 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1550 . 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  2496  eleq2  2497  clelab  2556  clabel  2557  nfel  2580  nfeld  2587  sbabel  2598  risset  2753  isset  2960  elex  2964  sbcabel  3238  ssel  3342  disjsn  3868  pwpw0  3946  pwsnALT  4010  mptpreima  5363  brfi1uzind  11715  ballotlem2  24746  eldm3  25385
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