Nuprl Lemma : coSet-equality2
∀x,y:coSet{i:l}.
((∀T:𝕌'. ((((a:Type × (a ⟶ T)) ⊆r T) ∧ (coSet{i:l} ⊆r T)) ⇒ (x = y ∈ T) ⇒ (x = y ∈ (a:Type × (a ⟶ T)))))
⇒ (x = y ∈ coSet{i:l}))
Proof
Definitions occuring in Statement :
coSet: coSet{i:l},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
cand: A c∧ B,
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
coSet-bisimulation: coSet-bisimulation{i:l}(x,y.R[x; y]),
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
uimplies: b supposing a,
true: True,
squash: ↓T
Lemmas referenced :
coSet_wf,
coSet-equality,
subtype_rel_self,
coSet-bisimulation_wf,
equal_wf,
istype-universe,
subtype_rel_wf,
subtype_rel_dep_function,
subtype_rel_product,
iff_weakening_equal,
coSet_subtype,
true_wf,
squash_wf,
subtype_coSet
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
independent_functionElimination,
productElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
universeEquality,
applyEquality,
productIsType,
independent_pairFormation,
sqequalRule,
because_Cache,
isectElimination,
instantiate,
productEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
cumulativity,
functionEquality,
functionIsType,
equalityIsType1,
rename,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
equalityIstype,
hypothesis_subsumption,
dependent_pairEquality_alt,
functionExtensionality
Latex:
\mforall{}x,y:coSet\{i:l\}.
((\mforall{}T:\mBbbU{}'. ((((a:Type \mtimes{} (a {}\mrightarrow{} T)) \msubseteq{}r T) \mwedge{} (coSet\{i:l\} \msubseteq{}r T)) {}\mRightarrow{} (x = y) {}\mRightarrow{} (x = y))) {}\mRightarrow{} (x = y))
Date html generated:
2019_10_31-AM-06_32_55
Last ObjectModification:
2018_12_13-PM-02_36_13
Theory : constructive!set!theory
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