Definition df-clab 2027

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 1393, which extends or "overloads" the wel 1394 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2033 and df-clel 2036, we introduce a new kind of variable (class variable) that can 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 1242 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2035 (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 2146 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 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".

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
df-clab	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

Detailed syntax breakdown of Definition df-clab

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2	1	cv 1242	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2026	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 1393	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1645	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 98	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2028  hbab1  2029  hbab  2031  cvjust  2035  abbi  2151  sb8ab  2159  cbvab  2160  clelab  2162  nfabd  2196  vjust  2558  dfsbcq2  2767  sbc8g  2771  csbabg  2907  unab  3204  inab  3205  difab  3206  rabeq0  3247  abeq0  3248  oprcl  3573  exss  3963  peano1  4317  peano2  4318  iotaeq  4875  nfvres  5206  abrexex2g  5747  opabex3d  5748  opabex3  5749  abrexex2  5751  bdab  9958  bdph  9970  bdcriota  10003