Nuprl Lemma : member-fset-intersection
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a,b:fset(T)]. ∀[x:T]. uiff(x ∈ a ⋂ b;x ∈ a ∧ x ∈ b)
Proof
Definitions occuring in Statement :
fset-intersection: a ⋂ b,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
fset-intersection: a ⋂ b,
guard: {T},
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],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
assert-deq-fset-member,
fset-member_witness,
and_wf,
fset-member_wf,
assert_wf,
deq-fset-member_wf,
assert_witness,
uiff_wf,
iff_weakening_uiff,
fset-filter_wf,
guard_wf,
member-fset-filter,
fset-intersection_wf,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_isectElimination,
hypothesis,
lemma_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
sqequalRule,
independent_pairEquality,
independent_functionElimination,
cumulativity,
lambdaEquality,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a,b:fset(T)]. \mforall{}[x:T]. uiff(x \mmember{} a \mcap{} b;x \mmember{} a \mwedge{} x \mmember{} b)
Date html generated:
2016_05_14-PM-03_40_04
Last ObjectModification:
2015_12_26-PM-06_41_27
Theory : finite!sets
Home
Index