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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 2107, which extends or "overloads" the wel 2108 definition connecting setvar variables, requires that both sides of ∈ be classes. In df-cleq 2770 and df-clel 2774, 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 1600 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2772 (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 2892 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 3467 which is used, for example, to convert elirrv 8792 to elirr 8793. 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 2764, df-cleq 2770, and df-clel 2774 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 1600 | . . 3 class 𝑥 |
3 | wph | . . . 4 wff 𝜑 | |
4 | vy | . . . 4 setvar 𝑦 | |
5 | 3, 4 | cab 2763 | . . 3 class {𝑦 ∣ 𝜑} |
6 | 2, 5 | wcel 2107 | . 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑} |
7 | 3, 4, 1 | wsb 2011 | . 2 wff [𝑥 / 𝑦]𝜑 |
8 | 6, 7 | wb 198 | 1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
This definition is referenced by: abid 2765 hbab1 2766 hbab 2768 cvjust 2772 cvjustOLD 2773 abbi2dv 2897 abbilem 2903 abbiiOLD 2909 cbvab 2913 cbvabv 2914 clelab 2916 nfabd 2954 nfabd2OLD 2956 vjust 3399 abv 3408 dfsbcq2 3655 sbc8g 3660 unab 4120 inab 4121 difab 4122 noel 4146 csbab 4234 exss 5165 iotaeq 6109 abrexex2g 7424 opabex3d 7425 opabex3 7426 bj-hbab1 33352 bj-abbi 33356 bj-vjust 33367 eliminable1 33419 bj-cleljustab 33426 bj-vexwt 33427 bj-vexwvt 33429 bj-ab0 33477 bj-snsetex 33527 bj-vjust2 33591 wl-clabv 33969 wl-clabtv 33971 wl-clabt 33972 wl-dfrabv 34000 wl-dfrabf 34002 cleljust2 38126 sn-vexwv 38127 cbvabvw 38142 |
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