Nuprl Lemma : strong-subtype-set1
∀[A:Type]. ∀[P,Q:A ⟶ ℙ]. strong-subtype({x:A| P[x]} ;{x:A| Q[x]} ) supposing ∀x:A. (P[x] ⇒ Q[x])
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {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,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q
Lemmas referenced :
strong-subtype-set,
strong-subtype-self,
strong-subtype_witness,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
universeEquality,
independent_functionElimination,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[P,Q:A {}\mrightarrow{} \mBbbP{}]. strong-subtype(\{x:A| P[x]\} ;\{x:A| Q[x]\} ) supposing \mforall{}x:A. (P[x] {}\mRightarrow{} Q[x])
Date html generated:
2016_05_13-PM-04_11_13
Last ObjectModification:
2015_12_26-AM-11_21_27
Theory : subtype_1
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