Nuprl Lemma : strong-subtype-member
∀[A,B:Type]. ∀[b:B]. ∀[a:A]. {b ∈ A supposing b = a ∈ B} supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
member: t ∈ T,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
strong-subtype_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lemma_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
isect_memberEquality,
because_Cache,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
lambdaEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[b:B]. \mforall{}[a:A]. \{b \mmember{} A supposing b = a\} supposing strong-subtype(A;B)
Date html generated:
2016_05_13-PM-04_11_39
Last ObjectModification:
2015_12_26-AM-11_12_25
Theory : subtype_1
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