Nuprl Lemma : member_not_nil
∀[T:Type]. ∀[L:T List]. ¬(L = [] ∈ (T List)) supposing ∃x:T. (x ∈ L)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
or: P ∨ Q,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
cons: [a / b],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
list-cases,
nil_member,
product_subtype_list,
cons_neq_nil,
equal-wf-T-base,
list_wf,
exists_wf,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_functionElimination,
unionElimination,
productElimination,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
sqequalRule,
cumulativity,
baseClosed,
lambdaEquality,
because_Cache,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mneg{}(L = []) supposing \mexists{}x:T. (x \mmember{} L)
Date html generated:
2017_04_14-AM-09_26_49
Last ObjectModification:
2017_02_27-PM-04_00_34
Theory : list_1
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