Nuprl Lemma : equipollent-implies-equal
∀[k,m:ℕ]. k = m ∈ ℤ supposing ℕk ~ ℕm
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
equipollent: A ~ B,
exists: ∃x:A. B[x],
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
biject: Bij(A;B;f)
Lemmas referenced :
injection_le,
inject_wf,
int_seg_wf,
equipollent_inversion,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equipollent_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
productElimination,
dependent_pairFormation_alt,
universeIsType,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[k,m:\mBbbN{}]. k = m supposing \mBbbN{}k \msim{} \mBbbN{}m
Date html generated:
2020_05_19-PM-10_00_23
Last ObjectModification:
2019_10_23-PM-02_49_32
Theory : equipollence!!cardinality!
Home
Index