Nuprl Lemma : singleton-subtype
∀[A,B:Type]. ∀[a:A]. ({z:B| z = a ∈ B} ⊆r A) supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
strong-subtype: strong-subtype(A;B),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cand: A c∧ B,
guard: {T},
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
subtype_rel_sets,
equal_wf,
exists_wf,
subtype_rel_transitivity,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
lambdaEquality,
cumulativity,
applyEquality,
hypothesis,
independent_isectElimination,
setElimination,
rename,
setEquality,
lambdaFormation,
dependent_pairFormation,
axiomEquality,
isect_memberEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[a:A]. (\{z:B| z = a\} \msubseteq{}r A) supposing strong-subtype(A;B)
Date html generated:
2017_04_14-AM-07_36_50
Last ObjectModification:
2017_02_27-PM-03_09_02
Theory : subtype_1
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