Nuprl Lemma : ccc-nset-remove1
∀K:Type. (CCCNSet(K) ⇒ (∀k0,k1:K. ((¬(k0 = k1 ∈ ℤ)) ⇒ CCCNSet({k:K| ¬(k = k0 ∈ ℤ)} ))))
Proof
Definitions occuring in Statement :
ccc-nset: CCCNSet(K),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
ge: i ≥ j ,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
contra-cc: CCC(T),
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
cand: A c∧ B,
so_apply: x[s],
uimplies: b supposing a,
nat: ℕ,
guard: {T},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
ccc-nset: CCCNSet(K),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
nat_properties,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
istype-universe,
ccc-nset_wf,
subtype_rel_self,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformnot_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
subtype_rel_transitivity,
istype-nat,
equal_wf,
not_wf,
nat_wf,
subtype_rel_set
Rules used in proof :
cumulativity,
promote_hyp,
equalityElimination,
unionElimination,
functionEquality,
universeEquality,
instantiate,
Error :productIsType,
Error :functionIsType,
Error :dependent_set_memberEquality_alt,
because_Cache,
Error :equalityIstype,
Error :universeIsType,
voidElimination,
Error :isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
Error :dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
Error :inhabitedIsType,
independent_isectElimination,
rename,
setElimination,
applyEquality,
intEquality,
Error :lambdaEquality_alt,
sqequalRule,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}K:Type. (CCCNSet(K) {}\mRightarrow{} (\mforall{}k0,k1:K. ((\mneg{}(k0 = k1)) {}\mRightarrow{} CCCNSet(\{k:K| \mneg{}(k = k0)\} ))))
Date html generated:
2019_06_20-PM-03_01_45
Last ObjectModification:
2019_06_13-PM-00_16_15
Theory : continuity
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