Nuprl Lemma : qle-iff
∀a,b:ℚ. uiff(a ≤ b;a < b ∨ (a = b ∈ ℚ))
Proof
Definitions occuring in Statement :
qle: r ≤ s
,
qless: r < s
,
rationals: ℚ
,
uiff: uiff(P;Q)
,
all: ∀x:A. B[x]
,
or: P ∨ Q
,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
member: t ∈ T
,
uall: ∀[x:A]. B[x]
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
uimplies: b supposing a
,
implies: P
⇒ Q
,
or: P ∨ Q
,
prop: ℙ
,
guard: {T}
,
false: False
Lemmas referenced :
qless_trichot_qorder,
qless_complement_qorder,
rationals_wf,
qle_witness,
equal_wf,
qless_wf,
qless_transitivity_2_qorder,
qless_irreflexivity,
qle_wf,
qle_weakening_lt_qorder,
qle_weakening_eq_qorder,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
because_Cache,
hypothesis,
productElimination,
independent_pairFormation,
isect_memberFormation,
introduction,
independent_functionElimination,
rename,
unionElimination,
inlFormation,
sqequalRule,
inrFormation,
independent_isectElimination,
voidElimination
Latex:
\mforall{}a,b:\mBbbQ{}. uiff(a \mleq{} b;a < b \mvee{} (a = b))
Date html generated:
2016_05_15-PM-10_45_48
Last ObjectModification:
2015_12_27-PM-07_53_14
Theory : rationals
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