Nuprl Lemma : fset-only_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:T ⟶ 𝔹]. ∀[s:fset(T)].
(only x ∈ s s.t. P[x] ∈ {x:T| x ∈ s ∧ (↑P[x])} ) supposing
((∀x,y:T. (x ∈ s ⇒ y ∈ s ⇒ (↑P[x]) ⇒ (↑P[y]) ⇒ (x = y ∈ T))) and
(¬(∀x:T. (x ∈ s ⇒ (¬↑P[x])))))
Proof
Definitions occuring in Statement :
fset-only: only x ∈ s s.t. P[x],
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
fset-only: only x ∈ s s.t. P[x],
and: P ∧ Q,
exists: ∃x:A. B[x],
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
guard: {T},
uiff: uiff(P;Q)
Lemmas referenced :
istype-universe,
fset-member_wf,
assert_wf,
not_wf,
all_wf,
fset_wf,
bool_wf,
deq_wf,
decidable__equal_int,
fset-size_wf,
fset-filter_wf,
fset-size-one,
iff_weakening_uiff,
member-fset-filter,
fset-item_wf,
fset-item-member,
and_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
Error :inhabitedIsType,
Error :universeIsType,
applyEquality,
Error :equalityIsType1,
Error :isect_memberEquality_alt,
Error :lambdaEquality_alt,
functionEquality,
because_Cache,
universeEquality,
lambdaEquality,
voidElimination,
lemma_by_obid,
independent_pairFormation,
dependent_pairFormation,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
Error :lambdaFormation_alt,
Error :productIsType,
productElimination,
Error :dependent_pairFormation_alt,
Error :isectIsType,
productEquality,
promote_hyp,
independent_isectElimination,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:fset(T)].
(only x \mmember{} s s.t. P[x] \mmember{} \{x:T| x \mmember{} s \mwedge{} (\muparrow{}P[x])\} ) supposing
((\mforall{}x,y:T. (x \mmember{} s {}\mRightarrow{} y \mmember{} s {}\mRightarrow{} (\muparrow{}P[x]) {}\mRightarrow{} (\muparrow{}P[y]) {}\mRightarrow{} (x = y))) and
(\mneg{}(\mforall{}x:T. (x \mmember{} s {}\mRightarrow{} (\mneg{}\muparrow{}P[x])))))
Date html generated:
2019_06_20-PM-02_00_16
Last ObjectModification:
2018_10_06-AM-11_23_46
Theory : finite!sets
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