Nuprl Lemma : rleq_antisymmetry
∀[x,y:ℝ]. (x = y) supposing ((y ≤ x) and (x ≤ y))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
req: 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,
implies: P ⇒ Q,
req: x = y,
all: ∀x:A. B[x],
prop: ℙ,
real: ℝ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
nat_plus: ℕ+,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
rleq-iff4,
nat_plus_wf,
req_witness,
rleq_wf,
real_wf,
absval_unfold,
subtract_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
because_Cache,
sqequalRule,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
lessCases,
sqequalAxiom,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[x,y:\mBbbR{}]. (x = y) supposing ((y \mleq{} x) and (x \mleq{} y))
Date html generated:
2017_10_03-AM-08_25_03
Last ObjectModification:
2017_07_28-AM-07_23_40
Theory : reals
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