Nuprl Lemma : fset-ac-le-singleton-empty
∀[T:Type]. ∀eq:EqDecider(T). ∀a:fset(fset(T)). fset-ac-le(eq;a;{{}})
Proof
Definitions occuring in Statement :
fset-ac-le: fset-ac-le(eq;ac1;ac2),
empty-fset: {},
fset-singleton: {x},
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
fset-null: fset-null(s),
null: null(as),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-singleton: {x},
cons: [a / b],
deq-f-subset: deq-f-subset(eq),
isl: isl(x),
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
decidable__assert,
empty-fset: {},
nil: [],
it: ⋅,
btrue: tt,
bfalse: ff,
true: True
Lemmas referenced :
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
decidable__assert,
fset_wf,
deq_wf,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
fset-singleton_wf,
empty-fset_wf,
fset-all-iff,
deq-fset_wf,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
assert_wf,
fset-member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
cumulativity,
because_Cache,
instantiate,
isectEquality,
universeEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
independent_functionElimination,
setElimination,
rename,
setEquality,
functionEquality,
productElimination,
independent_isectElimination,
natural_numberEquality
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}a:fset(fset(T)). fset-ac-le(eq;a;\{\{\}\})
Date html generated:
2016_05_14-PM-03_43_06
Last ObjectModification:
2015_12_26-PM-06_39_20
Theory : finite!sets
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