Nuprl Lemma : fset-absorption1
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a,b:fset(T)]. (a ⋃ a ⋂ b = a ∈ fset(T))
Proof
Definitions occuring in Statement :
fset-intersection: a ⋂ b,
fset-union: x ⋃ y,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
or: P ∨ Q,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
fset-extensionality,
fset-union_wf,
fset-intersection_wf,
fset-member_witness,
or_wf,
fset-member_wf,
and_wf,
member-fset-union,
member-fset-intersection,
uiff_wf,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
independent_pairFormation,
unionElimination,
because_Cache,
independent_functionElimination,
rename,
inlFormation,
addLevel,
dependent_functionElimination,
orFunctionality,
cumulativity,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[a,b:fset(T)]. (a \mcup{} a \mcap{} b = a)
Date html generated:
2016_05_14-PM-03_40_23
Last ObjectModification:
2015_12_26-PM-06_41_20
Theory : finite!sets
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