Nuprl Lemma : is-above-subtype
∀[A,B:Type]. ∀[a:A]. ∀[z:Base]. (is-above(A;a;z) ⇒ is-above(B;a;z)) supposing A ⊆r B
Proof
Definitions occuring in Statement :
is-above: is-above(T;a;z),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
base: Base,
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,
is-above: is-above(T;a;z),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
is-above_wf,
base_wf,
subtype_rel_wf,
equal_functionality_wrt_subtype_rel2,
equal-wf-base-T,
sqle_wf_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
universeEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
productEquality,
applyEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[a:A]. \mforall{}[z:Base]. (is-above(A;a;z) {}\mRightarrow{} is-above(B;a;z)) supposing A \msubseteq{}r B
Date html generated:
2016_05_13-PM-04_13_02
Last ObjectModification:
2015_12_26-AM-11_11_22
Theory : subtype_1
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