Nuprl Lemma : ss-mem-basic-whole
∀X:SeparationSpace. ∀x:Point(X). uiff(x ∈ ss-basic-whole();True)
Proof
Definitions occuring in Statement :
ss-basic-whole: ss-basic-whole(),
ss-mem-basic: x ∈ B,
ss-point: Point(ss),
separation-space: SeparationSpace,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
true: True
Definitions unfolded in proof :
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
true: True,
prop: ℙ,
uall: ∀[x:A]. B[x],
ss-basic-whole: ss-basic-whole(),
ss-mem-basic: x ∈ B,
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
select: L[n],
cons: [a / b],
ss-const: ss-const(c),
ss-ap: f(x),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
le: A ≤ B,
false: False,
not: ¬A,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
ss-mem-basic_wf,
ss-basic-whole_wf,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
rless-int,
int_seg_subtype,
false_wf,
int_seg_cases,
full-omega-unsat,
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,
int_seg_wf,
true_wf,
ss-point_wf,
separation-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
natural_numberEquality,
sqequalRule,
sqequalHypSubstitution,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
rename,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
hypothesis_subsumption,
addEquality,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality
Latex:
\mforall{}X:SeparationSpace. \mforall{}x:Point(X). uiff(x \mmember{} ss-basic-whole();True)
Date html generated:
2020_05_20-PM-01_22_24
Last ObjectModification:
2018_07_06-PM-02_22_13
Theory : intuitionistic!topology
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