Nuprl Lemma : ccc-nset-remove1

`∀K:Type. (CCCNSet(K) `` (∀k0,k1:K.  ((¬(k0 = k1 ∈ ℤ)) `` CCCNSet({k:K| ¬(k = k0 ∈ ℤ)} ))))`

Proof

Definitions occuring in Statement :  ccc-nset: `CCCNSet(K)` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` set: `{x:A| B[x]} ` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  nequal: `a ≠ b ∈ T ` assert: `↑b` bnot: `¬bb` sq_type: `SQType(T)` or: `P ∨ Q` bfalse: `ff` ge: `i ≥ j ` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` contra-cc: `CCC(T)` prop: `ℙ` top: `Top` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` not: `¬A` cand: `A c∧ B` so_apply: `x[s]` uimplies: `b supposing a` nat: `ℕ` guard: `{T}` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` and: `P ∧ Q` ccc-nset: `CCCNSet(K)` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  neg_assert_of_eq_int assert-bnot bool_subtype_base bool_wf subtype_base_sq bool_cases_sqequal eqff_to_assert nat_properties assert_of_eq_int eqtt_to_assert eq_int_wf istype-universe ccc-nset_wf subtype_rel_self int_formula_prop_wf int_formula_prop_not_lemma int_term_value_var_lemma int_formula_prop_eq_lemma istype-void int_formula_prop_and_lemma istype-int intformnot_wf itermVar_wf intformeq_wf intformand_wf full-omega-unsat subtype_rel_transitivity istype-nat equal_wf not_wf nat_wf subtype_rel_set
Rules used in proof :  cumulativity promote_hyp equalityElimination unionElimination functionEquality universeEquality instantiate Error :productIsType,  Error :functionIsType,  Error :dependent_set_memberEquality_alt,  because_Cache Error :equalityIstype,  Error :universeIsType,  voidElimination Error :isect_memberEquality_alt,  dependent_functionElimination int_eqEquality Error :dependent_pairFormation_alt,  independent_functionElimination approximateComputation natural_numberEquality equalitySymmetry equalityTransitivity Error :inhabitedIsType,  independent_isectElimination rename setElimination applyEquality intEquality Error :lambdaEquality_alt,  sqequalRule hypothesis hypothesisEquality isectElimination extract_by_obid introduction cut independent_pairFormation thin productElimination sqequalHypSubstitution Error :lambdaFormation_alt,  sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}K:Type.  (CCCNSet(K)  {}\mRightarrow{}  (\mforall{}k0,k1:K.    ((\mneg{}(k0  =  k1))  {}\mRightarrow{}  CCCNSet(\{k:K|  \mneg{}(k  =  k0)\}  ))))

Date html generated: 2019_06_20-PM-03_01_45
Last ObjectModification: 2019_06_13-PM-00_16_15

Theory : continuity

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