Nuprl Lemma : Peirce-subtype-dneg-elim
(∀[P,B:ℙ]. (((P ⇒ B) ⇒ P) ⇒ P)) ⊆r (∀[P:ℙ]. ((¬¬P) ⇒ P))
Proof
Definitions occuring in Statement :
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
so_apply: x[s],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
guard: {T},
false: False,
uimplies: b supposing a,
not: ¬A
Lemmas referenced :
uall_wf,
equal_wf,
isect_wf,
void_wf,
false_wf,
not_wf,
subtype_rel-equal
Rules used in proof :
hypothesisEquality,
functionEquality,
cumulativity,
thin,
isectElimination,
extract_by_obid,
introduction,
instantiate,
universeEquality,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
applyEquality,
cut,
isect_memberEquality,
lambdaEquality,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
because_Cache,
isectEquality,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
voidEquality,
voidElimination,
independent_isectElimination,
functionExtensionality
Latex:
(\mforall{}[P,B:\mBbbP{}]. (((P {}\mRightarrow{} B) {}\mRightarrow{} P) {}\mRightarrow{} P)) \msubseteq{}r (\mforall{}[P:\mBbbP{}]. ((\mneg{}\mneg{}P) {}\mRightarrow{} P))
Date html generated:
2018_05_21-PM-06_28_54
Last ObjectModification:
2017_12_27-PM-02_39_51
Theory : general
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