### Nuprl Lemma : equal-nat-inf-infinity

`∀[x:ℕ∞]. uiff(x = ∞ ∈ ℕ∞;∀i:ℕ. (¬(x = i∞ ∈ ℕ∞)))`

Proof

Definitions occuring in Statement :  nat-inf-infinity: `∞` nat2inf: `n∞` nat-inf: `ℕ∞` nat: `ℕ` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat-inf: `ℕ∞` nat-inf-infinity: `∞` nat: `ℕ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` bfalse: `ff` less_than: `a < b` nat2inf: `n∞` assert: `↑b` ifthenelse: `if b then t else f fi ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` true: `True` sq_type: `SQType(T)` bnot: `¬bb` subtract: `n - m` squash: `↓T`
Lemmas referenced :  equal_wf nat-inf_wf nat2inf_wf nat_wf equal-wf-T-base all_wf not_wf nat-inf-infinity-new nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf int_seg_wf int_seg_properties decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma decidable__equal_int int_seg_subtype false_wf intformeq_wf int_formula_prop_eq_lemma le_wf bool_wf decidable__lt lelt_wf itermAdd_wf int_term_value_add_lemma assert_wf iff_imp_equal_bool lt_int_wf true_wf assert_of_lt_int iff_wf add-zero decidable__equal_bool assert_elim btrue_neq_bfalse subtype_base_sq int_subtype_base eqtt_to_assert equal-wf-base eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot minus-one-mul add-commutes add-associates add-mul-special zero-mul zero-add squash_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation thin hypothesis sqequalHypSubstitution independent_functionElimination voidElimination extract_by_obid isectElimination hypothesisEquality sqequalRule lambdaEquality dependent_functionElimination because_Cache baseClosed productElimination independent_pairEquality isect_memberEquality equalityTransitivity equalitySymmetry axiomEquality hyp_replacement applyLambdaEquality setElimination rename dependent_set_memberEquality functionExtensionality intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality voidEquality computeAll unionElimination applyEquality hypothesis_subsumption equalityElimination addEquality functionEquality addLevel impliesFunctionality instantiate cumulativity promote_hyp imageElimination universeEquality equalityUniverse levelHypothesis imageMemberEquality

Latex:
\mforall{}[x:\mBbbN{}\minfty{}].  uiff(x  =  \minfty{};\mforall{}i:\mBbbN{}.  (\mneg{}(x  =  i\minfty{})))

Date html generated: 2017_10_01-AM-08_29_25
Last ObjectModification: 2017_07_26-PM-04_23_57

Theory : basic

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