Nuprl Lemma : subtype_by_equality
∀[A,B:Type]. ((∀x,y:Base. ((x = y ∈ A) ⇒ (x = y ∈ B))) ⇒ (A ⊆r B))
Proof
Definitions occuring in Statement :
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
label: ...$L... t,
guard: {T},
true: True
Lemmas referenced :
base_wf,
member_wf,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
Error :lambdaEquality_alt,
Error :universeIsType,
hypothesisEquality,
sqequalRule,
Error :functionIsType,
extract_by_obid,
hypothesis,
Error :inhabitedIsType,
Error :equalityIsType4,
because_Cache,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
axiomEquality,
Error :isect_memberEquality_alt,
isectElimination,
universeEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,B:Type]. ((\mforall{}x,y:Base. ((x = y) {}\mRightarrow{} (x = y))) {}\mRightarrow{} (A \msubseteq{}r B))
Date html generated:
2019_06_20-PM-00_28_09
Last ObjectModification:
2018_09_29-PM-11_18_30
Theory : subtype_1
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