| 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 2769 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2769 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 2114), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2763. Alternate definitions of 𝐴 ∈ 𝐵 (but that
require either 𝐴 or 𝐵 to be a set) are shown
by clel2 3542,
clel3 3544, and clel4 3545.
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.
While the three class definitions df-clab 2763, df-cleq 2769, and df-clel 2773
are eliminable and conservative and thus meet the requirements for sound
definitions, they are technically axioms in that they do not satisfy the
requirements for the current definition checker. The proofs of
conservativity require external justification that is beyond the scope
of the definition checker.
For a general discussion of the theory of classes, see
mmset.html#class. (Contributed by NM,
26-May-1993.) |
