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Definition df-clab 2608
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 1987, which extends or "overloads" the wel 1988 definition connecting setvar variables, requires that both sides of be classes. In df-cleq 2614 and df-clel 2617, 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 1479 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2616 (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 2729 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 3252 which is used, for example, to convert elirrv 8448 to elirr 8449.

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 2608, df-cleq 2614, and df-clel 2617 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 1479 . . 3 class 𝑥
3 wph . . . 4 wff 𝜑
4 vy . . . 4 setvar 𝑦
53, 4cab 2607 . . 3 class {𝑦𝜑}
62, 5wcel 1987 . 2 wff 𝑥 ∈ {𝑦𝜑}
73, 4, 1wsb 1877 . 2 wff [𝑥 / 𝑦]𝜑
86, 7wb 196 1 wff (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
Colors of variables: wff setvar class
This definition is referenced by:  abid  2609  hbab1  2610  hbab  2612  cvjust  2616  cbvab  2743  clelab  2745  nfabd2  2780  vjust  3187  abv  3192  dfsbcq2  3420  sbc8g  3425  unab  3870  inab  3871  difab  3872  csbab  3980  exss  4892  iotaeq  5818  abrexex2g  7090  opabex3d  7091  opabex3  7092  abrexex2  7094  bj-hbab1  32411  bj-abbi  32415  bj-vjust  32426  eliminable1  32482  bj-cleljustab  32489  bj-vexwt  32498  bj-vexwvt  32500  bj-ab0  32546  bj-snsetex  32595  bj-vjust2  32659  csbabgOLD  38530
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