Nuprl Lemma : isect_subtype_base
∀[A:Type]. ∀[B:A ⟶ Type]. (⋂a:A. B[a]) ⊆r Base supposing ∃a:A. (B[a] ⊆r Base)
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
isect_subtype_rel_trivial,
base_wf,
exists_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. (\mcap{}a:A. B[a]) \msubseteq{}r Base supposing \mexists{}a:A. (B[a] \msubseteq{}r Base)
Date html generated:
2016_05_13-PM-03_19_29
Last ObjectModification:
2015_12_26-AM-09_07_44
Theory : subtype_0
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