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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 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 {𝑦 ∣ 𝜑}. 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 {𝑦 ∣ 𝜑} 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.) |
Ref | Expression |
---|---|
df-clab | ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . 4 setvar 𝑥 | |
2 | 1 | cv 1473 | . . 3 class 𝑥 |
3 | wph | . . . 4 wff 𝜑 | |
4 | vy | . . . 4 setvar 𝑦 | |
5 | 3, 4 | cab 2595 | . . 3 class {𝑦 ∣ 𝜑} |
6 | 2, 5 | wcel 1976 | . 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑} |
7 | 3, 4, 1 | wsb 1866 | . 2 wff [𝑥 / 𝑦]𝜑 |
8 | 6, 7 | wb 194 | 1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [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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