Nuprl Lemma : rless-cases1
∀x,y:ℝ. ((x < y) ⇒ (∀z:ℝ. ((x < z) ∨ (z < y))))
Proof
Definitions occuring in Statement :
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
rsub: x - y,
iff: P ⇐⇒ Q
Lemmas referenced :
radd-positive-implies,
rsub_wf,
rless-implies-rless,
int-to-real_wf,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
req-iff-rsub-is-0,
real_wf,
rless_wf,
radd-ac,
radd-rminus-both,
radd-preserves-rless,
rminus_wf,
rless_functionality,
radd_comm,
radd_functionality,
req_weakening,
radd-rminus-assoc,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
inrFormation,
lemma_by_obid,
inlFormation,
unionElimination,
promote_hyp
Latex:
\mforall{}x,y:\mBbbR{}. ((x < y) {}\mRightarrow{} (\mforall{}z:\mBbbR{}. ((x < z) \mvee{} (z < y))))
Date html generated:
2017_10_03-AM-08_39_44
Last ObjectModification:
2017_07_28-AM-07_30_53
Theory : reals
Home
Index