Nuprl Lemma : nat2inf-one-one
∀[a,b:ℕ]. ((a∞ = b∞ ∈ ℕ∞)
⇒ (a = b ∈ ℕ))
Proof
Definitions occuring in Statement :
nat2inf: n∞
,
nat-inf: ℕ∞
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
implies: P
⇒ Q
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
implies: P
⇒ Q
,
nat2inf: n∞
,
nat-inf: ℕ∞
,
uimplies: b supposing a
,
sq_type: SQType(T)
,
all: ∀x:A. B[x]
,
guard: {T}
,
not: ¬A
,
nat: ℕ
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
prop: ℙ
,
ge: i ≥ j
,
decidable: Dec(P)
,
or: P ∨ Q
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
false: False
,
top: Top
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert_of_lt_int,
less_than_wf,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
le_wf,
equal_wf,
nat-inf_wf,
nat2inf_wf,
nat_wf,
assert_wf,
lt_int_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
applyLambdaEquality,
applyEquality,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
instantiate,
extract_by_obid,
isectElimination,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
promote_hyp,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
because_Cache,
axiomEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}[a,b:\mBbbN{}]. ((a\minfty{} = b\minfty{}) {}\mRightarrow{} (a = b))
Date html generated:
2017_10_01-AM-08_29_17
Last ObjectModification:
2017_07_26-PM-04_23_54
Theory : basic
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