Nuprl Lemma : bounded-above-strict
∀[A:Set(ℝ)]. (bounded-above(A) ⇒ (∃b:ℝ. A < b))
Proof
Definitions occuring in Statement :
bounded-above: bounded-above(A),
strict-upper-bound: A < b,
rset: Set(ℝ),
real: ℝ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
bounded-above: bounded-above(A),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rneq: x ≠ y,
guard: {T},
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
upper-bound: A ≤ b,
strict-upper-bound: A < b
Lemmas referenced :
real_wf,
upper-bound_wf,
rset_wf,
rless-int-fractions2,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
radd-preserves-rless,
int-to-real_wf,
rdiv_wf,
rless-int,
rless_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
rmul_wf,
rinv_wf2,
itermMultiply_wf,
strict-upper-bound_wf,
rless_functionality,
req_transitivity,
radd_functionality,
req_weakening,
rinv-as-rdiv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
rset-member_wf,
rless_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
productIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
natural_numberEquality,
dependent_set_memberEquality_alt,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
closedConclusion,
inrFormation_alt,
because_Cache,
int_eqEquality,
inhabitedIsType
Latex:
\mforall{}[A:Set(\mBbbR{})]. (bounded-above(A) {}\mRightarrow{} (\mexists{}b:\mBbbR{}. A < b))
Date html generated:
2019_10_29-AM-10_39_26
Last ObjectModification:
2019_04_19-PM-06_26_35
Theory : reals
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