Nuprl Lemma : i-member_functionality
∀I:Interval. ∀a,b:ℝ. a ∈ I ⇐⇒ b ∈ I supposing a = b
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
interval: Interval,
i-member: r ∈ I,
uimplies: b supposing a,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
guard: {T},
prop: ℙ,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
true: True
Lemmas referenced :
rleq_transitivity,
rleq_weakening,
req_inversion,
and_wf,
rleq_wf,
less_than'_wf,
rsub_wf,
real_wf,
nat_plus_wf,
req_wf,
req_witness,
rless_transitivity2,
rless_wf,
rless_transitivity1,
regular-int-seq_wf,
true_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
sqequalRule,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
because_Cache,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality
Latex:
\mforall{}I:Interval. \mforall{}a,b:\mBbbR{}. a \mmember{} I \mLeftarrow{}{}\mRightarrow{} b \mmember{} I supposing a = b
Date html generated:
2016_05_18-AM-08_19_22
Last ObjectModification:
2015_12_27-PM-11_58_41
Theory : reals
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