### Nuprl Lemma : l-ordered-from-upto-lt-nat-true

`∀[n,m:ℕ].  (l-ordered(ℕ;x,y.x < y;[n, m)) `⇐⇒` True)`

Proof

Definitions occuring in Statement :  l-ordered: `l-ordered(T;x,y.R[x; y];L)` from-upto: `[n, m)` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` true: `True`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` true: `True` prop: `ℙ` nat: `ℕ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` cand: `A c∧ B` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` rev_implies: `P `` Q` l-ordered: `l-ordered(T;x,y.R[x; y];L)`
Lemmas referenced :  l_before_wf member-less_than true_wf l-ordered-from-upto-lt-nat int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le nat_properties subtype_rel_sets less_than_wf le_wf and_wf subtype_rel_list from-upto_wf nat_wf l-ordered_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lambdaFormation natural_numberEquality lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesis setElimination rename hypothesisEquality applyEquality setEquality intEquality independent_isectElimination sqequalRule because_Cache lambdaEquality productElimination dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[n,m:\mBbbN{}].    (l-ordered(\mBbbN{};x,y.x  <  y;[n,  m))  \mLeftarrow{}{}\mRightarrow{}  True)

Date html generated: 2016_05_15-PM-04_37_31
Last ObjectModification: 2016_01_16-AM-11_18_45

Theory : general

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