Nuprl Lemma : f-subset-singleton
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[a:A]. ∀x:fset(A). uiff(x ⊆ {a};(x = {a} ∈ fset(A)) ∨ (x = {} ∈ fset(A)))
Proof
Definitions occuring in Statement :
empty-fset: {},
fset-singleton: {x},
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
f-subset: xs ⊆ ys,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
or: P ∨ Q,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
not: ¬A,
false: False,
rev_uimplies: rev_uimplies(P;Q),
top: Top
Lemmas referenced :
fset-member_wf,
fset-singleton_wf,
empty-fset_wf,
fset-member_witness,
iff_weakening_uiff,
equal_wf,
member-fset-singleton,
fset_wf,
deq_wf,
istype-universe,
decidable__assert,
fset-null_wf,
assert-fset-null,
fset-extensionality,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
decidable__fset-member,
mem_empty_lemma,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
independent_pairFormation,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
isectElimination,
axiomEquality,
hypothesis,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
rename,
functionIsType,
because_Cache,
isectIsType,
universeIsType,
extract_by_obid,
equalityIstype,
unionElimination,
unionIsType,
productElimination,
independent_functionElimination,
independent_isectElimination,
promote_hyp,
instantiate,
universeEquality,
inrFormation_alt,
equalityIsType1,
inlFormation_alt,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
hyp_replacement,
applyLambdaEquality,
voidElimination,
Error :memTop
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[a:A]. \mforall{}x:fset(A). uiff(x \msubseteq{} \{a\};(x = \{a\}) \mvee{} (x = \{\}))
Date html generated:
2020_05_19-PM-09_52_40
Last ObjectModification:
2020_01_04-PM-08_00_12
Theory : finite!sets
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