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Definition df-aiota 43292
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 43300); otherwise, it is not a set (see aiotaexb 43296), or even more concrete, it is the universe V (see aiotavb 43297). Since this is an alternative for df-iota 6316, 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 43296). 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 43280). Only the right to left implication would hold, see (negated) iotanul 6335. For defined cases, however, both definitions df-iota 6316 and df-aiota 43292 are equivalent, see reuaiotaiota 43295. (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 43290 . 2 class (℩'𝑥𝜑)
41, 2cab 2801 . . . . 5 class {𝑥𝜑}
5 vy . . . . . . 7 setvar 𝑦
65cv 1536 . . . . . 6 class 𝑦
76csn 4569 . . . . 5 class {𝑦}
84, 7wceq 1537 . . . 4 wff {𝑥𝜑} = {𝑦}
98, 5cab 2801 . . 3 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
109cint 4878 . 2 class {𝑦 ∣ {𝑥𝜑} = {𝑦}}
113, 10wceq 1537 1 wff (℩'𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
Colors of variables: wff setvar class
This definition is referenced by:  dfaiota2  43293  reuabaiotaiota  43294  aiotaexb  43296  aiotavb  43297
  Copyright terms: Public domain W3C validator