Nuprl Lemma : l_all_subtype
∀[A,B:Type]. ∀[L:A List]. ∀[P:B ⟶ ℙ]. (∀x∈L.P[x]) ⇒ (∀x∈L.P[x]) supposing A ⊆r B
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
l_all_wf,
l_member_wf,
subtype_rel_transitivity,
subtype_rel_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
setElimination,
setEquality,
because_Cache,
independent_isectElimination,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[L:A List]. \mforall{}[P:B {}\mrightarrow{} \mBbbP{}]. (\mforall{}x\mmember{}L.P[x]) {}\mRightarrow{} (\mforall{}x\mmember{}L.P[x]) supposing A \msubseteq{}r B
Date html generated:
2016_05_14-AM-07_49_52
Last ObjectModification:
2015_12_26-PM-04_45_42
Theory : list_1
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