Nuprl Lemma : rneq-rat2real
∀q1,q2:ℚ. uiff(rat2real(q1) ≠ rat2real(q2);¬(q1 = q2 ∈ ℚ))
Proof
Definitions occuring in Statement :
rat2real: rat2real(q),
rneq: x ≠ y,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
not: ¬A,
equal: s = t ∈ T,
rationals: ℚ
Definitions unfolded in proof :
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
false: False,
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
rneq: x ≠ y,
all: ∀x:A. B[x]
Lemmas referenced :
qless_irreflexivity,
qle_weakening_eq_qorder,
qless_transitivity_2_qorder,
qless_trichot_qorder,
rless-rat2real,
iff_weakening_uiff,
istype-void,
qless_wf,
rat2real_wf,
rless_wf,
rationals_wf
Rules used in proof :
equalitySymmetry,
rename,
promote_hyp,
independent_pairFormation,
functionIsTypeImplies,
voidElimination,
lambdaEquality_alt,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
inlFormation_alt,
unionElimination,
independent_isectElimination,
isect_memberFormation_alt,
productElimination,
equalityIstype,
functionIsType,
because_Cache,
unionIsType,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut,
universeIsType,
hypothesisEquality,
inhabitedIsType,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}q1,q2:\mBbbQ{}. uiff(rat2real(q1) \mneq{} rat2real(q2);\mneg{}(q1 = q2))
Date html generated:
2019_10_29-AM-09_59_44
Last ObjectModification:
2019_10_27-PM-02_39_55
Theory : reals
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