Nuprl Lemma : strict-upper-bound_functionality
∀A:Set(ℝ). ∀b,c:ℝ. {A < b ⇒ A < c} supposing b ≤ c
Proof
Definitions occuring in Statement :
strict-upper-bound: A < b,
rset: Set(ℝ),
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
strict-upper-bound: A < b,
guard: {T},
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
rge: x ≥ y
Lemmas referenced :
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
rset-member_wf,
all_wf,
rless_wf,
rleq_wf,
rset_wf,
rless_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}A:Set(\mBbbR{}). \mforall{}b,c:\mBbbR{}. \{A < b {}\mRightarrow{} A < c\} supposing b \mleq{} c
Date html generated:
2016_05_18-AM-08_09_17
Last ObjectModification:
2015_12_28-AM-01_15_41
Theory : reals
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