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Definition df-clab 2596
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 1976, which extends or "overloads" the wel 1977 definition connecting setvar variables, requires that both sides of be classes. In df-cleq 2602 and df-clel 2605, 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 2604 (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 2718 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 3238 which is used, for example, to convert elirrv 8364 to elirr 8365.

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 2596, df-cleq 2602, and df-clel 2605 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 2595 . . 3 class {𝑦𝜑}
62, 5wcel 1976 . 2 wff 𝑥 ∈ {𝑦𝜑}
73, 4, 1wsb 1866 . 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  2597  hbab1  2598  hbab  2600  cvjust  2604  cbvab  2732  clelab  2734  nfabd2  2769  vjust  3173  abv  3178  dfsbcq2  3404  sbc8g  3409  unab  3852  inab  3853  difab  3854  csbab  3959  exss  4852  iotaeq  5762  abrexex2g  7013  opabex3d  7014  opabex3  7015  abrexex2  7017  bj-hbab1  31765  bj-abbi  31769  bj-vjust  31780  eliminable1  31829  bj-vexwt  31844  bj-vexwvt  31846  bj-ab0  31890  bj-snsetex  31940  bj-vjust2  32002  csbabgOLD  37868
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