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
Home
Index