Nuprl Lemma : f-singleton-subset
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[a:A]. ∀[x:fset(A)]. uiff({a} ⊆ x;a ∈ x)
Proof
Definitions occuring in Statement :
fset-singleton: {x},
f-subset: xs ⊆ ys,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
f-subset: xs ⊆ ys,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T}
Lemmas referenced :
fset-member_witness,
all_wf,
isect_wf,
equal_wf,
fset-member_wf,
and_wf,
iff_weakening_uiff,
fset-singleton_wf,
member-fset-singleton,
uiff_wf,
f-subset_wf,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
independent_pairFormation,
isect_memberFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_functionElimination,
hypothesis,
cumulativity,
sqequalRule,
lambdaEquality,
lambdaFormation,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
productElimination,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
addLevel,
independent_isectElimination,
allFunctionality,
universeEquality,
independent_pairEquality
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[a:A]. \mforall{}[x:fset(A)]. uiff(\{a\} \msubseteq{} x;a \mmember{} x)
Date html generated:
2017_04_17-AM-09_19_02
Last ObjectModification:
2017_02_27-PM-05_22_24
Theory : finite!sets
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