Nuprl Lemma : classical-exists2
∀[T:Type]. ∀[P:T ⟶ ℙ]. uiff(¬(∀x:T. (¬P[x]));{∃x:T. P[x]})
Proof
Definitions occuring in Statement :
classical: {P},
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
classical: {P},
unit: Unit,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
or: P ∨ Q
Lemmas referenced :
it_wf,
classical-excluded-middle,
classical_wf,
all_wf,
not_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
axiomEquality,
natural_numberEquality,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
lambdaFormation,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
unionElimination,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. uiff(\mneg{}(\mforall{}x:T. (\mneg{}P[x]));\{\mexists{}x:T. P[x]\})
Date html generated:
2016_05_13-PM-03_17_01
Last ObjectModification:
2016_01_06-PM-05_20_32
Theory : core_2
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