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Definition df-clel 2477
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 2474 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2474 it does not strengthen the set of valid wffs of logic when the class variables are replaced with set variables (see cleljust 2105), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2468. Alternate definitions of (but that require either or to be a set) are shown by clel2 3122, clel3 3124, and clel4 3125.

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
Distinct variable groups:   ,   ,

Detailed syntax breakdown of Definition df-clel
StepHypRef Expression
1 cA . . 3
2 cB . . 3
31, 2wcel 1724 . 2
4 vx . . . . . 6
54cv 1661 . . . . 5
65, 1wceq 1662 . . . 4
75, 2wcel 1724 . . . 4
86, 7wa 360 . . 3
98, 4wex 1557 . 2
103, 9wb 178 1
Colors of variables: wff set class
This definition is referenced by:  eleq1  2541  eleq2  2542  clelab  2601  clabel  2602  nfel  2625  nfeld  2632  sbabel  2643  risset  2799  isset  3010  elex  3014  sbcabel  3300  ssel  3375  disjsn  3953  pwpw0  4031  pwsnALT  4096  mptpreima  5330  brfi1uzind  12006  ballotlem2  25512  eldm3  26215  bj-elsngl  30673
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