Nuprl Lemma : empty-fset-contains-none-of
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[cs:fset(fset(T))]. ↑fset-contains-none-of(eq;{};cs) supposing ¬{} ∈ cs
Proof
Definitions occuring in Statement :
fset-contains-none-of: fset-contains-none-of(eq;s;cs),
deq-fset: deq-fset(eq),
empty-fset: {},
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
iff: P ⇐⇒ Q
Lemmas referenced :
assert-fset-contains-none-of,
empty-fset_wf,
equal-wf-T-base,
fset_wf,
f-subset-empty,
f-subset_wf,
not_wf,
fset-member_wf,
deq-fset_wf,
fset-contains-none-of_wf,
deq_wf,
assert_witness,
and_wf,
equal_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
baseClosed,
addLevel,
impliesFunctionality,
because_Cache,
independent_functionElimination,
voidElimination,
universeEquality,
isect_memberFormation,
sqequalRule,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
dependent_set_memberEquality,
independent_pairFormation,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[cs:fset(fset(T))].
\muparrow{}fset-contains-none-of(eq;\{\};cs) supposing \mneg{}\{\} \mmember{} cs
Date html generated:
2016_10_21-AM-10_44_47
Last ObjectModification:
2016_07_12-AM-05_51_40
Theory : finite!sets
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