Metamath Proof Explorer


Theorem wel

Description: Extend wff definition to include atomic formulas with the membership predicate. This is read " x is an element of y ", " x is a member of y ", " x belongs to y ", or " y contains x ". Note: The phrase " y includes x " means " x is a subset of y "; to use it also for x e. y , as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactic construction introduces a binary non-logical predicate symbol e. (stylized lowercase epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel . This lets us avoid overloading the e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel is considered to be a primitive syntax, even though here it is artificially "derived" from wcel . Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program.) (Contributed by NM, 24-Jan-2006)

Ref Expression
Assertion wel x y