Nuprl Lemma : assert-fset-null
∀[T:Type]. ∀[s:fset(T)]. uiff(↑fset-null(s);s = {} ∈ fset(T))
Proof
Definitions occuring in Statement :
empty-fset: {},
fset-null: fset-null(s),
fset: fset(T),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
fset: fset(T),
all: ∀x:A. B[x],
quotient: x,y:A//B[x; y],
fset-null: fset-null(s),
empty-fset: {},
subtype_rel: A ⊆r B,
guard: {T},
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
assert: ↑b,
ifthenelse: if b then t else f fi ,
null: null(as),
nil: [],
it: ⋅,
btrue: tt
Lemmas referenced :
assert_wf,
fset-null_wf,
assert_witness,
equal-wf-T-base,
fset_wf,
list_wf,
set-equal_wf,
set-equal-reflex,
assert_of_null,
list_subtype_fset,
equal_functionality_wrt_subtype_rel2,
equal-wf-base,
equal_wf,
squash_wf,
true_wf,
quotient-member-eq,
set-equal-equiv,
empty-fset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
independent_functionElimination,
baseClosed,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
promote_hyp,
lambdaFormation,
pointwiseFunctionality,
pertypeElimination,
independent_isectElimination,
lambdaEquality,
productEquality,
dependent_functionElimination,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[s:fset(T)]. uiff(\muparrow{}fset-null(s);s = \{\})
Date html generated:
2017_04_17-AM-09_19_56
Last ObjectModification:
2017_02_27-PM-05_23_16
Theory : finite!sets
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