Nuprl Lemma : per-class-base-singleton
∀[T:Type]. ∀[a:T]. per-class(T;a) ≡ Base ⋂ {x:T| x = a ∈ T}
Proof
Definitions occuring in Statement :
per-class: per-class(T;a),
isect2: T1 ⋂ T2,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
isect2: T1 ⋂ T2,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
per-class: per-class(T;a),
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
cand: A c∧ B,
squash: ↓T
Lemmas referenced :
bool_wf,
eqtt_to_assert,
per-class_wf,
subtype_rel_b-union-right,
base_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
per-class-subtype-singleton,
isect2_decomp,
isect2_wf,
equal-wf-base-T
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
lambdaEquality,
isect_memberEquality,
hypothesisEquality,
applyEquality,
thin,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
because_Cache,
productElimination,
independent_isectElimination,
setElimination,
rename,
cumulativity,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry,
setEquality,
independent_pairEquality,
axiomEquality,
universeEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[a:T]. per-class(T;a) \mequiv{} Base \mcap{} \{x:T| x = a\}
Date html generated:
2017_04_14-AM-07_37_04
Last ObjectModification:
2017_02_27-PM-03_09_17
Theory : subtype_1
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