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Mirrors > Home > ILE Home > Th. List > Mathboxes > df-alsc | GIF version |
Description: Define "all some" applied to a class, which means 𝜑 is true for all 𝑥 in 𝐴 and there is at least one 𝑥 in 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) |
Ref | Expression |
---|---|
df-alsc | ⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cA | . . 3 class 𝐴 | |
4 | 1, 2, 3 | walsc 13687 | . 2 wff ∀!𝑥 ∈ 𝐴𝜑 |
5 | 1, 2, 3 | wral 2435 | . . 3 wff ∀𝑥 ∈ 𝐴 𝜑 |
6 | 2 | cv 1334 | . . . . 5 class 𝑥 |
7 | 6, 3 | wcel 2128 | . . . 4 wff 𝑥 ∈ 𝐴 |
8 | 7, 2 | wex 1472 | . . 3 wff ∃𝑥 𝑥 ∈ 𝐴 |
9 | 5, 8 | wa 103 | . 2 wff (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) |
10 | 4, 9 | wb 104 | 1 wff (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴)) |
Colors of variables: wff set class |
This definition is referenced by: alsconv 13690 alsc1d 13693 alsc2d 13694 |
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