Nuprl Lemma : MP+KS-imply-LEM

`(∀P:ℕ ⟶ ℙ. ((∀n:ℕ. Dec(P[n])) `` (¬(∀n:ℕ. (¬P[n]))) `` (∃n:ℕ. P[n])))`
` (∀A:ℙ. ∃a:ℕ ⟶ ℕ. (A `⇐⇒` ∃n:ℕ. ((a n) = 1 ∈ ℤ)))`
` (∀P:ℙ. (P ∨ (¬P)))`

Proof

Definitions occuring in Statement :  nat: `ℕ` decidable: `Dec(P)` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` implies: `P `` Q` or: `P ∨ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  guard: `{T}` false: `False` or: `P ∨ Q` not: `¬A` and: `P ∧ Q` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` exists: `∃x:A. B[x]` so_apply: `x[s]` nat: `ℕ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` prop: `ℙ` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q`
Lemmas referenced :  decidable__int_equal not-not-excluded-middle or_wf not_wf decidable_wf equal-wf-T-base iff_wf nat_wf exists_wf all_wf
Rules used in proof :  natural_numberEquality voidElimination promote_hyp because_Cache impliesFunctionality independent_functionElimination productElimination dependent_functionElimination baseClosed rename setElimination functionExtensionality applyEquality intEquality hypothesisEquality hypothesis functionEquality cumulativity lambdaEquality sqequalRule isectElimination sqequalHypSubstitution extract_by_obid introduction instantiate thin cut universeEquality lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
(\mforall{}P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}n:\mBbbN{}.  Dec(P[n]))  {}\mRightarrow{}  (\mneg{}(\mforall{}n:\mBbbN{}.  (\mneg{}P[n])))  {}\mRightarrow{}  (\mexists{}n:\mBbbN{}.  P[n])))
{}\mRightarrow{}  (\mforall{}A:\mBbbP{}.  \mexists{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  (A  \mLeftarrow{}{}\mRightarrow{}  \mexists{}n:\mBbbN{}.  ((a  n)  =  1)))
{}\mRightarrow{}  (\mforall{}P:\mBbbP{}.  (P  \mvee{}  (\mneg{}P)))

Date html generated: 2017_04_20-AM-07_36_03
Last ObjectModification: 2017_04_10-PM-05_57_57

Theory : continuity

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