Nuprl Lemma : inf-rless
∀[A:Set(ℝ)]. ∀b,c:ℝ. (inf(A) = b ⇒ (b < c ⇐⇒ ∃x:ℝ. ((x ∈ A) ∧ (x < c))))
Proof
Definitions occuring in Statement :
inf: inf(A) = b,
rset-member: x ∈ A,
rset: Set(ℝ),
rless: x < y,
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
inf: inf(A) = b,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
cand: A c∧ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
lower-bound: lower-bound(A;b),
guard: {T}
Lemmas referenced :
rless_wf,
rset-member_wf,
inf_wf,
real_wf,
rset_wf,
rsub_wf,
rless-implies-rless,
int-to-real_wf,
radd_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
itermAdd_wf,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
rless_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
productIsType,
inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
because_Cache,
independent_isectElimination,
dependent_pairFormation_alt,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}[A:Set(\mBbbR{})]. \mforall{}b,c:\mBbbR{}. (inf(A) = b {}\mRightarrow{} (b < c \mLeftarrow{}{}\mRightarrow{} \mexists{}x:\mBbbR{}. ((x \mmember{} A) \mwedge{} (x < c))))
Date html generated:
2019_10_29-AM-10_40_34
Last ObjectModification:
2019_04_22-PM-01_14_49
Theory : reals
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