Nuprl Lemma : qless-witness
∀[a,b:ℚ]. ⋅ ∈ a < b supposing a < b
Proof
Definitions occuring in Statement :
qless: r < s,
rationals: ℚ,
it: ⋅,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
prop: ℙ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable__qless,
bfalse: ff,
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
and: P ∧ Q,
uiff: uiff(P;Q),
guard: {T},
implies: P ⇒ Q,
sq_type: SQType(T),
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
assert: ↑b,
isl: isl(x),
true: True,
outl: outl(x),
false: False,
bnot: ¬bb,
outr: outr(x)
Lemmas referenced :
rationals_wf,
qless_wf,
decidable__qless,
subtype_rel_self,
all_wf,
decidable_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert-qpositive,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
qpositive_wf,
qsub_wf,
assert_wf,
bnot_wf,
not_wf,
int-subtype-rationals,
isl_wf
Rules used in proof :
inhabitedIsType,
because_Cache,
isect_memberEquality_alt,
hypothesisEquality,
thin,
isectElimination,
extract_by_obid,
universeIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyEquality,
instantiate,
functionEquality,
lambdaEquality,
impliesFunctionality,
lambdaFormation,
independent_pairFormation,
productElimination,
independent_functionElimination,
independent_isectElimination,
cumulativity,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
voidElimination
Latex:
\mforall{}[a,b:\mBbbQ{}]. \mcdot{} \mmember{} a < b supposing a < b
Date html generated:
2020_05_20-AM-09_15_52
Last ObjectModification:
2020_01_22-PM-05_16_17
Theory : rationals
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