Nuprl Lemma : decidable-subtype
∀[P,Q:ℙ]. (Dec(P) ⊆r Dec(Q)) supposing ((Q ⇒ P) and (P ⊆r Q))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
decidable: Dec(P),
subtype_rel: A ⊆r B,
or: P ∨ Q,
prop: ℙ,
implies: P ⇒ Q,
squash: ↓T
Lemmas referenced :
subtype_rel_wf,
or_wf,
subtype_rel_not,
not_wf,
subtype_rel_union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
lambdaFormation,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
axiomEquality,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[P,Q:\mBbbP{}]. (Dec(P) \msubseteq{}r Dec(Q)) supposing ((Q {}\mRightarrow{} P) and (P \msubseteq{}r Q))
Date html generated:
2016_05_13-PM-03_19_47
Last ObjectModification:
2016_01_14-PM-04_34_13
Theory : subtype_0
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