### Nuprl Lemma : injection_le

`∀[k,m:ℕ].  k ≤ m supposing ∃f:ℕk ⟶ ℕm. Inj(ℕk;ℕm;f)`

Proof

Definitions occuring in Statement :  inject: `Inj(A;B;f)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` exists: `∃x:A. B[x]` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` le: `A ≤ B` and: `P ∧ Q` not: `¬A` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_stable: `SqStable(P)` squash: `↓T` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` exists: `∃x:A. B[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` nat_plus: `ℕ+` less_than: `a < b` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` inject: `Inj(A;B;f)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality productElimination independent_pairEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality functionExtensionality applyEquality because_Cache imageMemberEquality baseClosed imageElimination unionElimination independent_pairFormation addEquality voidEquality intEquality minusEquality dependent_set_memberEquality applyLambdaEquality dependent_pairFormation sqequalIntensionalEquality promote_hyp multiplyEquality equalityElimination instantiate cumulativity impliesFunctionality

Latex:
\mforall{}[k,m:\mBbbN{}].    k  \mleq{}  m  supposing  \mexists{}f:\mBbbN{}k  {}\mrightarrow{}  \mBbbN{}m.  Inj(\mBbbN{}k;\mBbbN{}m;f)

Date html generated: 2017_04_14-AM-07_33_32
Last ObjectModification: 2017_02_27-PM-03_11_31

Theory : fun_1

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