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Definition df-prv 32881
Description: Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀m 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.)
Assertion
Ref Expression
df-prv ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚m ω)}
Distinct variable group:   𝑢,𝑚

Detailed syntax breakdown of Definition df-prv
StepHypRef Expression
1 cprv 32874 . 2 class
2 vm . . . . . 6 setvar 𝑚
32cv 1541 . . . . 5 class 𝑚
4 vu . . . . . 6 setvar 𝑢
54cv 1541 . . . . 5 class 𝑢
6 csate 32873 . . . . 5 class Sat
73, 5, 6co 7172 . . . 4 class (𝑚 Sat 𝑢)
8 com 7601 . . . . 5 class ω
9 cmap 8439 . . . . 5 class m
103, 8, 9co 7172 . . . 4 class (𝑚m ω)
117, 10wceq 1542 . . 3 wff (𝑚 Sat 𝑢) = (𝑚m ω)
1211, 2, 4copab 5092 . 2 class {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚m ω)}
131, 12wceq 1542 1 wff ⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚m ω)}
Colors of variables: wff setvar class
This definition is referenced by:  prv  32963
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