### Nuprl Lemma : cardinality-le_functionality

`∀[T:Type]. ∀n:ℕ+. ∀[m:ℕ]. {|T| ≤ n `` |T| ≤ m} supposing n ≤ m`

Proof

Definitions occuring in Statement :  cardinality-le: `|T| ≤ n` nat_plus: `ℕ+` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` guard: `{T}` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` le: `A ≤ B` and: `P ∧ Q` not: `¬A` implies: `P `` Q` false: `False` nat: `ℕ` nat_plus: `ℕ+` prop: `ℙ` guard: `{T}` cardinality-le: `|T| ≤ n` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` lelt: `i ≤ j < k` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` less_than': `less_than'(a;b)` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` surject: `Surj(A;B;f)`
Lemmas referenced :  less_than'_wf cardinality-le_wf nat_plus_subtype_nat le_wf nat_wf nat_plus_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int int_seg_wf lelt_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf false_wf int_seg_properties nat_properties nat_plus_properties decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf surject_wf intformle_wf int_formula_prop_le_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality lambdaEquality dependent_functionElimination hypothesisEquality voidElimination extract_by_obid isectElimination setElimination rename hypothesis axiomEquality equalityTransitivity equalitySymmetry applyEquality universeEquality dependent_pairFormation because_Cache unionElimination equalityElimination independent_isectElimination functionExtensionality natural_numberEquality dependent_set_memberEquality independent_pairFormation promote_hyp instantiate cumulativity independent_functionElimination approximateComputation int_eqEquality intEquality isect_memberEquality voidEquality

Latex:
\mforall{}[T:Type].  \mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}[m:\mBbbN{}].  \{|T|  \mleq{}  n  {}\mRightarrow{}  |T|  \mleq{}  m\}  supposing  n  \mleq{}  m

Date html generated: 2018_05_21-PM-00_39_32
Last ObjectModification: 2018_05_19-AM-06_45_12

Theory : list_1

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