Nuprl Lemma : rlessw_wf
∀x:ℝ. ∀y:{y:ℝ| x < y} . (rlessw(x;y) ∈ x < y)
Proof
Definitions occuring in Statement :
rlessw: rlessw(x;y),
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rlessw: rlessw(x;y),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
real: ℝ,
int_upper: {i...},
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
top: Top,
exists: ∃x:A. B[x],
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
rev_uimplies: rev_uimplies(P;Q),
rless: x < y,
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
add-swap,
add-associates,
assert_wf,
all_wf,
less_than_transitivity1,
assert_of_lt_int,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
le_wf,
subtype_rel_sets,
int_upper_wf,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
false_wf,
decidable__lt,
lt_int_wf,
less_than_wf,
quick-find_wf,
rless_wf,
real_wf,
set_wf,
rless-iff4
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
setElimination,
thin,
rename,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
introduction,
imageMemberEquality,
baseClosed,
addEquality,
unionElimination,
voidElimination,
independent_isectElimination,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
dependent_pairFormation,
setEquality,
int_eqEquality,
computeAll
Latex:
\mforall{}x:\mBbbR{}. \mforall{}y:\{y:\mBbbR{}| x < y\} . (rlessw(x;y) \mmember{} x < y)
Date html generated:
2016_05_18-AM-07_04_19
Last ObjectModification:
2016_01_17-AM-01_50_41
Theory : reals
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