Nuprl Lemma : l_exists_cons
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀x:T. ∀L:T List. ((∃y∈[x / L]. P[y]) ⇐⇒ P[x] ∨ (∃y∈L. P[y]))
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
cons: [a / b],
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
cand: A c∧ B,
true: True
Lemmas referenced :
l_member_wf,
cons_member,
cons_wf,
l_exists_iff,
l_exists_wf,
subtype_rel_self,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
Error :productIsType,
Error :inhabitedIsType,
hypothesisEquality,
Error :unionIsType,
Error :equalityIsType1,
Error :universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
applyEquality,
because_Cache,
independent_functionElimination,
Error :dependent_pairFormation_alt,
dependent_functionElimination,
promote_hyp,
Error :lambdaEquality_alt,
setElimination,
rename,
functionExtensionality,
cumulativity,
Error :setIsType,
unionElimination,
Error :inlFormation_alt,
Error :inrFormation_alt,
instantiate,
universeEquality,
Error :functionIsType,
hyp_replacement,
equalitySymmetry,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
natural_numberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}x:T. \mforall{}L:T List. ((\mexists{}y\mmember{}[x / L]. P[y]) \mLeftarrow{}{}\mRightarrow{} P[x] \mvee{} (\mexists{}y\mmember{}L. P[y]))
Date html generated:
2019_06_20-PM-00_41_13
Last ObjectModification:
2018_10_02-PM-06_05_00
Theory : list_0
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