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Definition df-clel 2282
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 2279 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2279 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 1961), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2273. Alternate definitions of  A  e.  B (but that require either  A or  B to be a set) are shown by clel2 2907, clel3 2909, and clel4 2910.

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/mpegif/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 1687 . 2  wff  A  e.  B
4 vx . . . . . 6  set  x
54cv 1624 . . . . 5  class  x
65, 1wceq 1625 . . . 4  wff  x  =  A
75, 2wcel 1687 . . . 4  wff  x  e.  B
86, 7wa 360 . . 3  wff  ( x  =  A  /\  x  e.  B )
98, 4wex 1530 . 2  wff  E. x
( x  =  A  /\  x  e.  B
)
103, 9wb 178 1  wff  ( A  e.  B  <->  E. x
( x  =  A  /\  x  e.  B
) )
Colors of variables: wff set class
This definition is referenced by:  eleq1  2346  eleq2  2347  clelab  2406  clabel  2407  nfel  2430  nfeld  2437  sbabel  2448  risset  2593  isset  2795  elex  2799  sbcabel  3071  ssel  3177  disjsn  3696  pwpw0  3766  pwsnALT  3825  mptpreima  5166  ballotlem2  23042
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