Nuprl Lemma : nil_member-variant
∀[T,A:Type]. ∀x:T. (x ∈ []) ⇐⇒ False supposing A ⊆r T
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nil: [],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
false: False,
universe: Type
Definitions unfolded in proof :
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
false: False,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
decidable: Dec(P),
or: P ∨ Q,
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
length_of_nil_lemma,
stuck-spread,
base_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
exists_wf,
nat_wf,
less_than_wf,
length_wf,
nil_wf,
equal_wf,
select_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
false_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
independent_pairFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
hypothesisEquality,
setElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
computeAll,
productEquality,
because_Cache,
cumulativity,
applyEquality,
unionElimination,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}x:T. (x \mmember{} []) \mLeftarrow{}{}\mRightarrow{} False supposing A \msubseteq{}r T
Date html generated:
2018_05_21-PM-06_33_17
Last ObjectModification:
2017_07_26-PM-04_52_09
Theory : general
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