Nuprl Lemma : strong-subtype-eq1
∀[A,B:Type]. ∀[b:B]. ∀[a:A]. ((b ∈ A) c∧ (b = a ∈ A)) supposing ((b = a ∈ B) and strong-subtype(A;B))
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
member: t ∈ T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cand: A c∧ B,
implies: P ⇒ Q,
guard: {T},
strong-subtype: strong-subtype(A;B),
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
all: ∀x:A. B[x]
Lemmas referenced :
strong-subtype-implies,
equal_wf,
exists_wf,
strong-subtype_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
productElimination,
dependent_pairFormation,
hypothesisEquality,
applyEquality,
sqequalRule,
dependent_set_memberEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[b:B]. \mforall{}[a:A]. ((b \mmember{} A) c\mwedge{} (b = a)) supposing ((b = a) and strong-subtype(A;B))
Date html generated:
2016_05_13-PM-04_11_31
Last ObjectModification:
2015_12_26-AM-11_21_22
Theory : subtype_1
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