Nuprl Lemma : cantor-theorem-on-power-set-prop
∀[T:Type]. ∀f:T ⟶ T ⟶ ℙ. ∃P:T ⟶ ℙ. ∀x:T. (¬(∀y:T. (P y ⇐⇒ f x y)))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
not_wf,
all_wf,
iff_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
hypothesis,
sqequalRule,
independent_functionElimination,
voidElimination,
functionEquality,
universeEquality,
dependent_functionElimination,
productElimination,
instantiate,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mexists{}P:T {}\mrightarrow{} \mBbbP{}. \mforall{}x:T. (\mneg{}(\mforall{}y:T. (P y \mLeftarrow{}{}\mRightarrow{} f x y)))
Date html generated:
2018_05_21-PM-08_36_13
Last ObjectModification:
2017_07_26-PM-06_00_46
Theory : general
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