Nuprl Lemma : equipollent-product-zero
∀[A:Type]. (A × ℕ0 ~ ℕ0 ∧ ℕ0 × A ~ ℕ0)
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
and: P ∧ Q,
product: x:A × B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
equipollent: A ~ B,
exists: ∃x:A. B[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
iff: P ⇐⇒ Q
Lemmas referenced :
ext-eq_weakening,
equipollent_weakening_ext-eq,
equipollent-product-com,
equipollent_functionality_wrt_equipollent,
equal_wf,
biject_wf,
int_seg_wf,
lelt_wf,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
independent_pairFormation,
hypothesis,
universeEquality,
dependent_pairFormation,
lambdaEquality,
dependent_set_memberEquality,
productElimination,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
setElimination,
rename,
independent_isectElimination,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
productEquality,
cumulativity,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
applyEquality,
setEquality,
independent_functionElimination
Latex:
\mforall{}[A:Type]. (A \mtimes{} \mBbbN{}0 \msim{} \mBbbN{}0 \mwedge{} \mBbbN{}0 \mtimes{} A \msim{} \mBbbN{}0)
Date html generated:
2016_05_14-PM-04_01_07
Last ObjectModification:
2016_01_14-PM-11_06_34
Theory : equipollence!!cardinality!
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