Nuprl Lemma : rless-case_wf
∀[x,y:ℝ]. ∀[n:x < y]. ∀[z:ℝ]. (rless-case(x;y;n;z) ∈ (x < z) ∨ (z < y))
Proof
Definitions occuring in Statement :
rless-case: rless-case(x;y;n;z),
rless: x < y,
real: ℝ,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rless-cases,
or: P ∨ Q,
prop: ℙ,
uimplies: b supposing a,
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
real: ℝ,
so_apply: x[s],
nat_plus: ℕ+,
rless-case: rless-case(x;y;n;z),
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
rless-cases,
subtype_base_sq,
or_wf,
rless_wf,
union_subtype_base,
set_subtype_base,
nat_plus_wf,
less_than_wf,
int_subtype_base,
real_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
addEquality,
applyEquality,
setElimination,
rename,
natural_numberEquality,
intEquality,
because_Cache,
functionExtensionality,
functionEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[n:x < y]. \mforall{}[z:\mBbbR{}]. (rless-case(x;y;n;z) \mmember{} (x < z) \mvee{} (z < y))
Date html generated:
2018_05_22-PM-01_46_13
Last ObjectModification:
2017_10_26-PM-10_54_06
Theory : reals
Home
Index