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Mirrors > Home > MPE Home > Th. List > df-clab | Structured version Visualization version GIF version |
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 "{𝑦 ∣ 𝜑} " is read "the class of
all sets 𝑦
such that 𝜑(𝑦) is true." We do not define
{𝑦 ∣
𝜑} 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 {𝑦 ∣ 𝜑}. 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 {𝑦 ∣ 𝜑} 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.) |
Ref | Expression |
---|---|
df-clab | ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . 4 setvar 𝑥 | |
2 | 1 | cv 1479 | . . 3 class 𝑥 |
3 | wph | . . . 4 wff 𝜑 | |
4 | vy | . . . 4 setvar 𝑦 | |
5 | 3, 4 | cab 2607 | . . 3 class {𝑦 ∣ 𝜑} |
6 | 2, 5 | wcel 1987 | . 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑} |
7 | 3, 4, 1 | wsb 1877 | . 2 wff [𝑥 / 𝑦]𝜑 |
8 | 6, 7 | wb 196 | 1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [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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