### Nuprl Lemma : disjoint-iff-null-intersection

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[a,b:T List].  uiff(l_disjoint(T;a;b);l_intersection(eq;a;b) = [] ∈ (T List))`

Proof

Definitions occuring in Statement :  l_intersection: `l_intersection(eq;L1;L2)` l_disjoint: `l_disjoint(T;l1;l2)` nil: `[]` list: `T List` deq: `EqDecider(T)` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  l_disjoint: `l_disjoint(T;l1;l2)` uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` false: `False` or: `P ∨ Q` cons: `[a / b]` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B`
Lemmas referenced :  all_wf not_wf l_member_wf equal-wf-T-base list_wf l_intersection_wf deq_wf list-cases product_subtype_list equal_wf cons_member member-intersection null_nil_lemma btrue_wf member-implies-null-eq-bfalse btrue_neq_bfalse
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut independent_pairFormation hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality lambdaEquality productEquality lambdaFormation independent_functionElimination voidElimination dependent_functionElimination because_Cache baseClosed productElimination independent_pairEquality isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry universeEquality unionElimination promote_hyp hypothesis_subsumption rename inlFormation hyp_replacement applyLambdaEquality independent_isectElimination

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[a,b:T  List].    uiff(l\_disjoint(T;a;b);l\_intersection(eq;a;b)  =  [])

Date html generated: 2017_04_17-AM-09_16_20
Last ObjectModification: 2017_02_27-PM-05_21_36

Theory : decidable!equality

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