Nuprl Lemma : member-fset-list-union
∀[T:Type]. ∀eq:EqDecider(T). ∀ss:fset(T) List. ∀x:T. (x ∈ fset-list-union(eq;ss) ⇐⇒ (∃s∈ss. x ∈ s))
Proof
Definitions occuring in Statement :
fset-list-union: fset-list-union(eq;ss),
fset-member: a ∈ s,
fset: fset(T),
l_exists: (∃x∈L. P[x]),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
fset-list-union: fset-list-union(eq;ss),
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
false: False,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
guard: {T}
Lemmas referenced :
list_induction,
fset_wf,
all_wf,
iff_wf,
fset-member_wf,
fset-list-union_wf,
l_exists_wf,
l_member_wf,
list_wf,
reduce_nil_lemma,
mem_empty_lemma,
false_wf,
l_exists_nil,
l_exists_wf_nil,
reduce_cons_lemma,
or_wf,
member-fset-union,
l_exists_cons,
fset-union_wf,
cons_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
introduction,
independent_pairFormation,
because_Cache,
productElimination,
independent_pairEquality,
addLevel,
allFunctionality,
impliesFunctionality,
applyEquality,
unionElimination,
inlFormation,
inrFormation,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}ss:fset(T) List. \mforall{}x:T. (x \mmember{} fset-list-union(eq;ss) \mLeftarrow{}{}\mRightarrow{} (\mexists{}s\mmember{}ss. x \mmember{} s))
Date html generated:
2016_05_14-PM-03_40_34
Last ObjectModification:
2015_12_26-PM-06_41_01
Theory : finite!sets
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