Nuprl Lemma : equipollent-powerset
∀n:ℕ. ∀T:Type. (T ~ ℕn ⇒ powerset(T) ~ ℕ2^n)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
powerset: powerset(T),
equipollent: A ~ B,
exp: i^n,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
powerset: powerset(T),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equipollent_wf,
int_seg_wf,
nat_wf,
exp_wf2,
equipollent-exp,
false_wf,
le_wf,
equipollent_functionality_wrt_equipollent,
function_functionality_wrt_equipollent_left,
equipollent_weakening_ext-eq,
ext-eq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
universeEquality,
functionEquality,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
sqequalRule,
independent_pairFormation,
independent_functionElimination,
independent_isectElimination,
productElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}T:Type. (T \msim{} \mBbbN{}n {}\mRightarrow{} powerset(T) \msim{} \mBbbN{}2\^{}n)
Date html generated:
2016_05_14-PM-04_02_36
Last ObjectModification:
2015_12_26-PM-07_42_46
Theory : equipollence!!cardinality!
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