Nuprl Lemma : l_exists_append
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀L1,L2:T List. ((∃x∈L1 @ L2. P[x]) ⇐⇒ (∃x∈L1. P[x]) ∨ (∃x∈L2. P[x]))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
append: as @ bs,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
cand: A c∧ B
Lemmas referenced :
list_wf,
member_append,
l_member_wf,
append_wf,
subtype_rel_self,
l_exists_wf,
l_exists_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
Error :functionIsType,
universeEquality,
independent_pairFormation,
productElimination,
Error :dependent_pairFormation_alt,
dependent_functionElimination,
independent_functionElimination,
promote_hyp,
sqequalRule,
Error :productIsType,
Error :unionIsType,
applyEquality,
because_Cache,
unionElimination,
Error :inlFormation_alt,
Error :inrFormation_alt,
instantiate,
Error :lambdaEquality_alt,
setElimination,
rename,
functionExtensionality,
cumulativity,
Error :setIsType
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}L1,L2:T List. ((\mexists{}x\mmember{}L1 @ L2. P[x]) \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L1. P[x]) \mvee{} (\mexists{}x\mmember{}L2. P[x]))
Date html generated:
2019_06_20-PM-00_41_32
Last ObjectModification:
2018_10_02-AM-10_06_10
Theory : list_0
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