Nuprl Lemma : not_over_exists
∀[T:Type]. ∀[Q:T ⟶ ℙ]. uiff(¬(∃x:T. Q[x]);∀x:T. (¬Q[x]))
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False
Lemmas referenced :
not_wf,
exists_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
applyEquality,
hypothesisEquality,
cut,
hypothesis,
thin,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
universeEquality,
because_Cache,
Error :universeIsType,
introduction,
extract_by_obid,
isectElimination,
Error :functionIsType,
functionEquality,
cumulativity,
Error :isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. uiff(\mneg{}(\mexists{}x:T. Q[x]);\mforall{}x:T. (\mneg{}Q[x]))
Date html generated:
2019_06_20-AM-11_16_33
Last ObjectModification:
2018_09_26-AM-10_24_17
Theory : core_2
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