Nuprl Lemma : equipollent-zero
∀[A:Type]. (A ~ ℕ0 ⇐⇒ ¬A)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
equipollent: A ~ B,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f)
Lemmas referenced :
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
not_wf,
biject_wf,
int_seg_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
cut,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
hypothesisEquality,
lemma_by_obid,
isectElimination,
functionEquality,
natural_numberEquality,
lambdaEquality,
universeEquality,
productElimination,
rename,
introduction,
applyEquality,
because_Cache,
setElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}[A:Type]. (A \msim{} \mBbbN{}0 \mLeftarrow{}{}\mRightarrow{} \mneg{}A)
Date html generated:
2016_05_14-PM-04_01_46
Last ObjectModification:
2016_01_14-PM-11_06_10
Theory : equipollence!!cardinality!
Home
Index