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Definition df-clel 2431
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 2428 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2428 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2097), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2422. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 3064, clel3 3066, and clel4 3067.

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  2495  eleq2  2496  clelab  2555  clabel  2556  nfel  2579  nfeld  2586  sbabel  2597  risset  2745  isset  2952  elex  2956  sbcabel  3230  ssel  3334  disjsn  3860  pwpw0  3938  pwsnALT  4002  mptpreima  5355  brfi1uzind  11707  ballotlem2  24738  eldm3  25377
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