Nuprl Lemma : f-subset-empty
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:fset(T)]. (x ⊆ {} ⇐⇒ x = {} ∈ fset(T))
Proof
Definitions occuring in Statement :
empty-fset: {},
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
f-subset: xs ⊆ ys,
all: ∀x:A. B[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
guard: {T},
top: Top,
false: False
Lemmas referenced :
f-subset_wf,
empty-fset_wf,
equal-wf-T-base,
fset_wf,
fset-member_witness,
fset-member_wf,
deq_wf,
fset-extensionality,
mem_empty_lemma,
f-subset_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
baseClosed,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
universeEquality,
independent_isectElimination,
voidElimination,
voidEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:fset(T)]. (x \msubseteq{} \{\} \mLeftarrow{}{}\mRightarrow{} x = \{\})
Date html generated:
2016_10_21-AM-10_44_21
Last ObjectModification:
2016_07_12-AM-05_51_25
Theory : finite!sets
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