Nuprl Lemma : fset-size-one
∀[T:Type]. ∀eq:EqDecider(T). ∀s:fset(T). (||s|| = 1 ∈ ℤ ⇐⇒ ∃x:T. (x ∈ s ∧ (∀y:T. y = x ∈ T supposing y ∈ s)))
Proof
Definitions occuring in Statement :
fset-size: ||s||,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
prop: ℙ,
cand: A c∧ B,
fset: fset(T),
quotient: x,y:A//B[x; y],
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
fset-member: a ∈ s,
fset-size: ||s||,
guard: {T},
sq_type: SQType(T),
set-equal: set-equal(T;x;y)
Lemmas referenced :
istype-int,
fset-size_wf,
set_subtype_base,
le_wf,
int_subtype_base,
fset-member_wf,
fset_wf,
deq_wf,
istype-universe,
fset-item_wf,
fset-item-member,
list_wf,
set-equal_wf,
set-equal-reflex,
equal_wf,
squash_wf,
true_wf,
equal-wf-base,
quotient-member-eq,
set-equal-equiv,
subtype_rel_self,
assert-deq-member,
remove-repeats-length-one,
iff_weakening_equal,
l_member_wf,
subtype_base_sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :equalityIstype,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
intEquality,
Error :lambdaEquality_alt,
closedConclusion,
natural_numberEquality,
because_Cache,
independent_isectElimination,
baseClosed,
sqequalBase,
equalitySymmetry,
Error :productIsType,
Error :universeIsType,
Error :functionIsType,
Error :isectIsType,
instantiate,
universeEquality,
Error :dependent_pairFormation_alt,
equalityTransitivity,
Error :inhabitedIsType,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
productElimination,
Error :equalityIsType4,
dependent_functionElimination,
imageElimination,
independent_functionElimination,
imageMemberEquality,
cumulativity
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}s:fset(T). (||s|| = 1 \mLeftarrow{}{}\mRightarrow{} \mexists{}x:T. (x \mmember{} s \mwedge{} (\mforall{}y:T. y = x supposing y \mmember{} s)))
Date html generated:
2019_06_20-PM-02_00_11
Last ObjectModification:
2018_11_22-AM-10_00_33
Theory : finite!sets
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