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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-prv | Structured version Visualization version GIF version |
Description: Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀 ↑m ω), the interpretation of the wff using the membership relation on 𝑀 is true. Since ⊧ is defined in terms of the interpretations making the given formula true, it is not defined on the empty "model" 𝑀 = ∅, since there are no interpretations. In particular, the empty set on the LHS of ⊧ should not be interpreted as the empty model. Statement prv0 32965 shows that our definition yields ∅⊧𝑈 for all formulas, though of course the formula ∃𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Ref | Expression |
---|---|
df-prv | ⊢ ⊧ = {〈𝑚, 𝑢〉 ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cprv 32874 | . 2 class ⊧ | |
2 | vm | . . . . . 6 setvar 𝑚 | |
3 | 2 | cv 1541 | . . . . 5 class 𝑚 |
4 | vu | . . . . . 6 setvar 𝑢 | |
5 | 4 | cv 1541 | . . . . 5 class 𝑢 |
6 | csate 32873 | . . . . 5 class Sat∈ | |
7 | 3, 5, 6 | co 7172 | . . . 4 class (𝑚 Sat∈ 𝑢) |
8 | com 7601 | . . . . 5 class ω | |
9 | cmap 8439 | . . . . 5 class ↑m | |
10 | 3, 8, 9 | co 7172 | . . . 4 class (𝑚 ↑m ω) |
11 | 7, 10 | wceq 1542 | . . 3 wff (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω) |
12 | 11, 2, 4 | copab 5092 | . 2 class {〈𝑚, 𝑢〉 ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)} |
13 | 1, 12 | wceq 1542 | 1 wff ⊧ = {〈𝑚, 𝑢〉 ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)} |
Colors of variables: wff setvar class |
This definition is referenced by: prv 32963 |
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