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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 2127, which extends or "overloads" the wel 2128 definition connecting setvar variables, requires that both sides of ∈ be classes. In df-cleq 2741 and df-clel 2744, 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 1619 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2743 (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 2858 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 3394 which is used, for example, to convert elirrv 8654 to elirr 8655. 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 2735, df-cleq 2741, and df-clel 2744 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 1619 | . . 3 class 𝑥 |
3 | wph | . . . 4 wff 𝜑 | |
4 | vy | . . . 4 setvar 𝑦 | |
5 | 3, 4 | cab 2734 | . . 3 class {𝑦 ∣ 𝜑} |
6 | 2, 5 | wcel 2127 | . 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑} |
7 | 3, 4, 1 | wsb 2034 | . 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 2736 hbab1 2737 hbab 2739 cvjust 2743 cbvab 2872 clelab 2874 nfabd2 2910 vjust 3329 abv 3334 dfsbcq2 3567 sbc8g 3572 unab 4025 inab 4026 difab 4027 csbab 4139 exss 5068 iotaeq 6008 abrexex2g 7297 opabex3d 7298 opabex3 7299 abrexex2OLD 7303 bj-hbab1 33048 bj-abbi 33052 bj-vjust 33063 eliminable1 33117 bj-cleljustab 33124 bj-vexwt 33131 bj-vexwvt 33133 bj-ab0 33179 bj-snsetex 33228 bj-vjust2 33292 csbabgOLD 39521 |
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