Nuprl Lemma : cantor-theorem-on-power-set
∀[T:Type]. (¬T ~ powerset(T))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
powerset: powerset(T),
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
equipollent: A ~ B,
exists: ∃x:A. B[x],
biject: Bij(A;B;f),
and: P ∧ Q,
surject: Surj(A;B;f),
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
assert: ↑b,
powerset: powerset(T)
Lemmas referenced :
equipollent_wf,
powerset_wf,
bool_wf,
bnot_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
eqtt_to_assert,
btrue_neq_bfalse,
equal-wf-base,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_seg_wf,
equipollent_functionality_wrt_equipollent2,
function_functionality_wrt_equipollent_right,
equipollent-two
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
because_Cache,
universeEquality,
functionEquality,
productElimination,
applyEquality,
functionExtensionality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate
Latex:
\mforall{}[T:Type]. (\mneg{}T \msim{} powerset(T))
Date html generated:
2018_05_21-PM-08_36_26
Last ObjectModification:
2017_07_26-PM-06_00_58
Theory : general
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