Nuprl Lemma : equipollent_interval
∀a,b:ℤ. {a..b-} ~ ℕb - a
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
all: ∀x:A. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
equipollent: A ~ B,
member: t ∈ T,
exists: ∃x:A. B[x],
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
guard: {T},
surject: Surj(A;B;f),
uiff: uiff(P;Q)
Lemmas referenced :
add-subtract-cancel,
add-member-int_seg2,
equal_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
biject_wf,
int_seg_wf,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
subtract_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
intEquality,
dependent_pairFormation,
lambdaEquality,
dependent_set_memberEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_pairFormation,
productElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality
Latex:
\mforall{}a,b:\mBbbZ{}. \{a..b\msupminus{}\} \msim{} \mBbbN{}b - a
Date html generated:
2016_05_14-PM-04_01_16
Last ObjectModification:
2016_01_14-PM-11_06_37
Theory : equipollence!!cardinality!
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