Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-aiota Structured version   Visualization version   GIF version

Definition df-aiota 44277
Description: Alternate version of Russell's definition of a description binder, which can be read as "the unique 𝑥 such that 𝜑", where 𝜑 ordinarily contains 𝑥 as a free variable. Our definition is meaningful only when there is exactly one 𝑥 such that 𝜑 is true (see aiotaval 44287); otherwise, it is not a set (see aiotaexb 44281), or even more concrete, it is the universe V (see aiotavb 44282). Since this is an alternative for df-iota 6355, we call this symbol ℩' alternate iota in the following.

The advantage of this definition is the clear distinguishability of the defined and undefined cases: the alternate iota over a wff is defined iff it is a set (see aiotaexb 44281). With the original definition, there is no corresponding theorem (∃!𝑥𝜑 ↔ (℩𝑥𝜑) ≠ ∅), because can be a valid unique set satisfying a wff (see, for example, iota0def 44232). Only the right to left implication would hold, see (negated) iotanul 6375. For defined cases, however, both definitions df-iota 6355 and df-aiota 44277 are equivalent, see reuaiotaiota 44280. (Proposed by BJ, 13-Aug-2022.) (Contributed by AV, 24-Aug-2022.)

Assertion
Ref Expression
df-aiota (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Detailed syntax breakdown of Definition df-aiota
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
31, 2caiota 44275 . 2 class (℩'𝑥𝜑)
41, 2cab 2715 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1542 . . . . . 6 class 𝑦
76csn 4555 . . . . 5 class {𝑦}
84, 7wceq 1543 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2715 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cint 4873 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1543 1 wff (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfaiota2  44278  reuabaiotaiota  44279  aiotaexb  44281  aiotavb  44282
  Copyright terms: Public domain W3C validator