Nuprl Lemma : not-not-all-int_seg-xmiddle

`∀a,b:ℤ. ∀P:{a..b-} ⟶ ℙ.  (¬¬(∀i:{a..b-}. (P[i] ∨ (¬P[i]))))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` not: `¬A` or: `P ∨ Q` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` and: `P ∧ Q` ge: `i ≥ j ` le: `A ≤ B` cand: `A c∧ B` less_than: `a < b` squash: `↓T` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` not: `¬A` int_seg: `{i..j-}` sq_stable: `SqStable(P)` lelt: `i ≤ j < k` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` less_than': `less_than'(a;b)` true: `True` or: `P ∨ Q` so_apply: `x[s]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` decidable: `Dec(P)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nat_plus: `ℕ+` sq_type: `SQType(T)`
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination independent_pairFormation productElimination imageElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination universeIsType sqequalRule lambdaEquality_alt dependent_functionElimination functionIsTypeImplies inhabitedIsType addEquality isect_memberEquality_alt minusEquality imageMemberEquality baseClosed functionIsType unionIsType applyEquality instantiate universeEquality because_Cache dependent_set_memberEquality_alt baseApply closedConclusion unionElimination productIsType multiplyEquality functionEquality unionEquality cumulativity intEquality equalityTransitivity equalitySymmetry equalityIstype

Latex:
\mforall{}a,b:\mBbbZ{}.  \mforall{}P:\{a..b\msupminus{}\}  {}\mrightarrow{}  \mBbbP{}.    (\mneg{}\mneg{}(\mforall{}i:\{a..b\msupminus{}\}.  (P[i]  \mvee{}  (\mneg{}P[i]))))

Date html generated: 2020_05_19-PM-09_36_10
Last ObjectModification: 2019_10_17-PM-02_49_36

Theory : int_1

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