Nuprl Lemma : fset-minimals-singleton
∀[T:Type]
∀eq:EqDecider(T). ∀x:fset(T). (fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys); {x}) = {x} ∈ fset(fset(T)))
Proof
Definitions occuring in Statement :
fset-minimals: fset-minimals(x,y.less[x; y]; s),
f-proper-subset-dec: f-proper-subset-dec(eq;xs;ys),
fset-singleton: {x},
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
fset-all: fset-all(s;x.P[x]),
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
f-proper-subset: xs ⊆≠ ys,
false: False
Lemmas referenced :
fset_wf,
deq_wf,
deq-fset_wf,
fset-minimals_wf,
f-proper-subset-dec_wf,
fset-singleton_wf,
equal_wf,
fset-all_wf,
bnot_wf,
assert_witness,
fset-null_wf,
fset-filter_wf,
fset-member_wf,
uiff_wf,
fset-extensionality,
fset-member_witness,
iff_transitivity,
iff_weakening_uiff,
member-fset-minimals,
member-fset-singleton,
fset-all-iff,
assert_wf,
not_wf,
f-proper-subset_wf,
assert_of_bnot,
assert-f-proper-subset-dec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
because_Cache,
universeEquality,
independent_pairFormation,
productElimination,
productEquality,
independent_pairEquality,
independent_functionElimination,
independent_isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
addLevel,
impliesFunctionality,
hyp_replacement,
applyLambdaEquality,
voidElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}x:fset(T). (fset-minimals(xs,ys.f-proper-subset-dec(eq;xs;ys); \{x\}) = \{x\})
Date html generated:
2017_04_17-AM-09_23_35
Last ObjectModification:
2017_02_27-PM-05_25_03
Theory : finite!sets
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