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Mirrors > Home > ILE Home > Th. List > df-womni | GIF version |
Description: A weakly omniscient set
is one where we can decide whether a predicate
(here represented by a function 𝑓) holds (is equal to 1o) for
all elements or not. Generalization of definition 2.4 of [Pierik],
p. 9.
In particular, ω ∈ WOmni is known as the Weak Limited Principle of Omniscience (WLPO). The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.) |
Ref | Expression |
---|---|
df-womni | ⊢ WOmni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cwomni 7085 | . 2 class WOmni | |
2 | vy | . . . . . . 7 setvar 𝑦 | |
3 | 2 | cv 1331 | . . . . . 6 class 𝑦 |
4 | c2o 6347 | . . . . . 6 class 2o | |
5 | vf | . . . . . . 7 setvar 𝑓 | |
6 | 5 | cv 1331 | . . . . . 6 class 𝑓 |
7 | 3, 4, 6 | wf 5159 | . . . . 5 wff 𝑓:𝑦⟶2o |
8 | vx | . . . . . . . . . 10 setvar 𝑥 | |
9 | 8 | cv 1331 | . . . . . . . . 9 class 𝑥 |
10 | 9, 6 | cfv 5163 | . . . . . . . 8 class (𝑓‘𝑥) |
11 | c1o 6346 | . . . . . . . 8 class 1o | |
12 | 10, 11 | wceq 1332 | . . . . . . 7 wff (𝑓‘𝑥) = 1o |
13 | 12, 8, 3 | wral 2432 | . . . . . 6 wff ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o |
14 | 13 | wdc 820 | . . . . 5 wff DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o |
15 | 7, 14 | wi 4 | . . . 4 wff (𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o) |
16 | 15, 5 | wal 1330 | . . 3 wff ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o) |
17 | 16, 2 | cab 2140 | . 2 class {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)} |
18 | 1, 17 | wceq 1332 | 1 wff WOmni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)} |
Colors of variables: wff set class |
This definition is referenced by: iswomni 7087 |
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