Nuprl Lemma : t-sqle-subtype
∀[A,B:Type]. ∀[a,b:A]. (t-sqle(A;a;b) ⇒ t-sqle(B;a;b)) supposing A ⊆r B
Proof
Definitions occuring in Statement :
t-sqle: t-sqle(T;a;b),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
t-sqle: t-sqle(T;a;b),
squash: ↓T,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
per-class: per-class(T;a),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
subtype_rel_wf,
t-sqle_wf,
b-union_wf,
subtype_rel_transitivity,
base_wf,
subtype_rel_b-union-right,
per-class_wf,
exists_wf,
sqle_wf_base,
subtype_rel_per-class
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
sqequalRule,
setElimination,
rename,
cumulativity,
lambdaEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[a,b:A]. (t-sqle(A;a;b) {}\mRightarrow{} t-sqle(B;a;b)) supposing A \msubseteq{}r B
Date html generated:
2016_05_13-PM-04_12_52
Last ObjectModification:
2016_01_14-PM-07_29_00
Theory : subtype_1
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