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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 1977, which extends or "overloads" the wel 1978 definition connecting setvar variables, requires that both sides of ∈ be classes. In df-cleq 2603 and df-clel 2606, 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 1474 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2605 (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 2719 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 3239 which is used, for example, to convert elirrv 8387 to elirr 8388. 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 2597, df-cleq 2603, and df-clel 2606 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 1474 | . . 3 class 𝑥 |
3 | wph | . . . 4 wff 𝜑 | |
4 | vy | . . . 4 setvar 𝑦 | |
5 | 3, 4 | cab 2596 | . . 3 class {𝑦 ∣ 𝜑} |
6 | 2, 5 | wcel 1977 | . 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑} |
7 | 3, 4, 1 | wsb 1867 | . 2 wff [𝑥 / 𝑦]𝜑 |
8 | 6, 7 | wb 195 | 1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
This definition is referenced by: abid 2598 hbab1 2599 hbab 2601 cvjust 2605 cbvab 2733 clelab 2735 nfabd2 2770 vjust 3174 abv 3179 dfsbcq2 3405 sbc8g 3410 unab 3853 inab 3854 difab 3855 csbab 3960 exss 4858 iotaeq 5776 abrexex2g 7036 opabex3d 7037 opabex3 7038 abrexex2 7040 bj-hbab1 31959 bj-abbi 31963 bj-vjust 31974 eliminable1 32033 bj-vexwt 32048 bj-vexwvt 32050 bj-ab0 32094 bj-snsetex 32144 bj-vjust2 32206 csbabgOLD 38072 |
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