Nuprl Lemma : rleq_weakening_rless
∀[x,y:ℝ]. x ≤ y supposing x < y
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
real: ℝ,
decidable: Dec(P),
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
nat_plus: ℕ+,
prop: ℙ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T
Lemmas referenced :
rless_transitivity,
rless_wf,
real_wf,
rsub_wf,
less_than'_wf,
nat_plus_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
less_than_wf,
decidable__lt,
nat_plus_properties,
decidable__le,
rleq-iff4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
applyEquality,
setElimination,
rename,
addEquality,
natural_numberEquality,
hypothesis,
unionElimination,
dependent_set_memberFormation,
dependent_set_memberEquality,
because_Cache,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_pairEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination
Latex:
\mforall{}[x,y:\mBbbR{}]. x \mleq{} y supposing x < y
Date html generated:
2016_05_18-AM-07_06_15
Last ObjectModification:
2016_01_17-AM-01_51_14
Theory : reals
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