MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-clab Structured version   Visualization version   GIF version

Definition df-clab 2501
Description: Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. 𝑥 and 𝑦 need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, 𝜑 will have 𝑦 as a free variable, and "{𝑦𝜑} " is read "the class of all sets 𝑦 such that 𝜑(𝑦) is true." We do not define {𝑦𝜑} in isolation but only as part of an expression that extends or "overloads" the relationship.

This is our first use of the symbol to connect classes instead of sets. The syntax definition wcel 1938, which extends or "overloads" the wel 1939 definition connecting setvar variables, requires that both sides of be classes. In df-cleq 2507 and df-clel 2510, we introduce a new kind of variable (class variable) that can be substituted with expressions such as {𝑦𝜑}. In the present definition, the 𝑥 on the left-hand side is a setvar variable. Syntax definition cv 1473 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2509 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2623 for a quick overview).

Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is done with theorems such as vtoclg 3143 which is used, for example, to convert elirrv 8262 to elirr 8263.

This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction {𝑦𝜑} a "class term".

While the three class definitions df-clab 2501, df-cleq 2507, and df-clel 2510 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.)

Assertion
Ref Expression
df-clab (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)

Detailed syntax breakdown of Definition df-clab
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
21cv 1473 . . 3 class 𝑥
3 wph . . . 4 wff 𝜑
4 vy . . . 4 setvar 𝑦
53, 4cab 2500 . . 3 class {𝑦𝜑}
62, 5wcel 1938 . 2 wff 𝑥 ∈ {𝑦𝜑}
73, 4, 1wsb 1830 . 2 wff [𝑥 / 𝑦]𝜑
86, 7wb 194 1 wff (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
Colors of variables: wff setvar class
This definition is referenced by:  abid  2502  hbab1  2503  hbab  2505  cvjust  2509  cbvab  2637  clelab  2639  nfabd2  2674  vjust  3078  abv  3083  dfsbcq2  3309  sbc8g  3314  unab  3756  inab  3757  difab  3758  csbab  3863  exss  4756  iotaeq  5661  abrexex2g  6910  opabex3d  6911  opabex3  6912  abrexex2  6914  bj-hbab1  31801  bj-abbi  31805  bj-vjust  31816  eliminable1  31865  bj-vexwt  31880  bj-vexwvt  31882  bj-ab0  31926  bj-snsetex  31976  bj-vjust2  32038  csbabgOLD  37954
  Copyright terms: Public domain W3C validator