**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 2030 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2030 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 1810), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2024.
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.) |