Nuprl Lemma : member-intersection
∀[A:Type]. ∀eq:EqDecider(A). ∀L1,L2:A List. ∀x:A. ((x ∈ l_intersection(eq;L1;L2)) ⇐⇒ (x ∈ L1) ∧ (x ∈ L2))
Proof
Definitions occuring in Statement :
l_intersection: l_intersection(eq;L1;L2),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
l_intersection: l_intersection(eq;L1;L2),
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
assert-deq-member,
assert_wf,
deq-member_wf,
iff_wf,
and_wf,
l_member_wf,
member_filter,
filter_wf5,
list_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
sqequalRule,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
impliesFunctionality,
hypothesis,
lemma_by_obid,
isectElimination,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
andLevelFunctionality,
lambdaEquality,
cumulativity,
setElimination,
rename,
setEquality,
universeEquality
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A). \mforall{}L1,L2:A List. \mforall{}x:A. ((x \mmember{} l\_intersection(eq;L1;L2)) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L1) \mwedge{} (x \mmember{} L2))
Date html generated:
2016_05_14-PM-03_32_30
Last ObjectModification:
2015_12_26-PM-06_01_25
Theory : decidable!equality
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