Nuprl Lemma : isect_subtype_rel_trivial
∀[A,C:Type]. ∀[B:A ⟶ Type]. (⋂x:A. B[x]) ⊆r C supposing ∃x:A. (B[x] ⊆r C)
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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x]
Lemmas referenced :
exists_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
productElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
isectElimination,
equalityTransitivity,
equalitySymmetry,
hypothesis,
isectEquality,
axiomEquality,
lemma_by_obid,
isect_memberEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A,C:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. (\mcap{}x:A. B[x]) \msubseteq{}r C supposing \mexists{}x:A. (B[x] \msubseteq{}r C)
Date html generated:
2016_05_13-PM-03_18_58
Last ObjectModification:
2015_12_26-AM-09_08_00
Theory : subtype_0
Home
Index