Nuprl Lemma : sup-unique
∀[A:Set(ℝ)]. ∀[b,c:ℝ]. (b = c) supposing (sup(A) = c and sup(A) = b)
Proof
Definitions occuring in Statement :
sup: sup(A) = b,
rset: Set(ℝ),
req: x = y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
sup: sup(A) = b,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
top: Top,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
prop: ℙ,
cand: A c∧ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
upper-bound: A ≤ b,
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
rgt: x > y,
rsub: x - y
Lemmas referenced :
rless_functionality,
radd-rminus-assoc,
radd-assoc,
rless_transitivity1,
radd-preserves-rleq,
radd-zero-both,
radd_comm,
rminus-rminus,
rmul-zero-both,
radd-int,
rminus-as-rmul,
rmul_functionality,
rmul-distrib2,
rmul-identity1,
req_inversion,
req_transitivity,
radd-ac,
req_weakening,
rminus-radd,
radd_functionality,
rleq_functionality,
uiff_transitivity,
rleq_weakening_rless,
rsub_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rmul_wf,
radd_wf,
rleq_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
rless-int-fractions2,
rset_wf,
rset-member_wf,
exists_wf,
real_wf,
all_wf,
upper-bound_wf,
and_wf,
req_witness,
nat_plus_wf,
rabs_wf,
less_than'_wf,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
rminus_wf,
rsub_wf,
rmax_lb,
rabs-as-rmax,
infinitesmal-difference
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
natural_numberEquality,
setElimination,
rename,
inrFormation,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
independent_pairEquality,
applyEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
multiplyEquality,
addEquality,
promote_hyp
Latex:
\mforall{}[A:Set(\mBbbR{})]. \mforall{}[b,c:\mBbbR{}]. (b = c) supposing (sup(A) = c and sup(A) = b)
Date html generated:
2016_05_18-AM-08_10_20
Last ObjectModification:
2016_01_17-AM-02_28_15
Theory : reals
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