### Nuprl Lemma : biject-int-nat

`∃f:ℤ ⟶ ℕ. Bij(ℤ;ℕ;f)`

Proof

Definitions occuring in Statement :  biject: `Bij(A;B;f)` nat: `ℕ` exists: `∃x:A. B[x]` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  exists: `∃x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` prop: `ℙ` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` biject: `Bij(A;B;f)` inject: `Inj(A;B;f)` subtype_rel: `A ⊆r B` surject: `Surj(A;B;f)` ge: `i ≥ j ` true: `True` nequal: `a ≠ b ∈ T ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_nzero: `ℤ-o` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)`
Lemmas referenced :  le_int_wf eqtt_to_assert assert_of_le_int decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermMultiply_wf itermVar_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_mul_lemma int_term_value_var_lemma int_formula_prop_wf le_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf subtract_wf itermAdd_wf itermSubtract_wf itermMinus_wf int_term_value_add_lemma int_term_value_subtract_lemma int_term_value_minus_lemma nat_wf int_subtype_base biject_wf nat_properties decidable__equal_int intformeq_wf int_formula_prop_eq_lemma eq_int_wf assert_of_eq_int neg_assert_of_eq_int set_subtype_base div_rem_sum nequal_wf rem_bounds_1 less_than_wf add-is-int-iff multiply-is-int-iff false_wf intformless_wf int_formula_prop_less_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  cut natural_numberEquality hypothesisEquality introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesis Error :inhabitedIsType,  Error :lambdaFormation_alt,  unionElimination equalityElimination productElimination independent_isectElimination sqequalRule Error :dependent_set_memberEquality_alt,  dependent_functionElimination multiplyEquality approximateComputation independent_functionElimination int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :universeIsType,  equalityTransitivity equalitySymmetry Error :equalityIsType1,  promote_hyp instantiate cumulativity addEquality minusEquality Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality intEquality applyLambdaEquality Error :equalityIsType2,  remainderEquality setElimination rename divideEquality imageMemberEquality pointwiseFunctionality imageElimination

Latex:
\mexists{}f:\mBbbZ{}  {}\mrightarrow{}  \mBbbN{}.  Bij(\mBbbZ{};\mBbbN{};f)

Date html generated: 2019_06_20-PM-01_18_59
Last ObjectModification: 2018_10_06-PM-06_09_25

Theory : int_2

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