Nuprl Lemma : is-above-axiom-general
∀[T:Type]. ∀[z:Base]. z ~ Ax supposing is-above(T;Ax;z) supposing T ⊆r Base
Proof
Definitions occuring in Statement :
is-above: is-above(T;a;z),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
base: Base,
universe: Type,
sqequal: s ~ t,
axiom: Ax
Definitions unfolded in proof :
is-above: is-above(T;a;z),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
and: P ∧ Q,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
has-value: (a)↓,
or: P ∨ Q,
not: ¬A,
false: False
Lemmas referenced :
subtype_base_sq,
base_wf,
subtype_rel_self,
istype-sqle,
istype-base,
subtype_rel_wf,
istype-universe,
has-value-monotonic,
has-value_wf_base,
is-exception_wf,
has-value-implies-dec-isaxiom-2,
not-btrue-sqle-bfalse
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
productElimination,
hypothesisEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomSqEquality,
Error :productIsType,
Error :inhabitedIsType,
Error :equalityIstype,
Error :universeIsType,
baseClosed,
sqequalBase,
because_Cache,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
universeEquality,
divergentSqle,
sqleReflexivity,
unionElimination,
sqleRule,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[z:Base]. z \msim{} Ax supposing is-above(T;Ax;z) supposing T \msubseteq{}r Base
Date html generated:
2019_06_20-PM-00_28_15
Last ObjectModification:
2019_01_20-PM-02_27_27
Theory : subtype_1
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