Nuprl Lemma : equipollent-product-one
∀[A:Type]. (A × ℕ1 ~ A ∧ ℕ1 × A ~ A)
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 :
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
all: ∀x:A. B[x],
uimplies: b supposing a,
true: True,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
cand: A c∧ B
Lemmas referenced :
top_wf,
int_seg_wf,
equipollent_wf,
equipollent_functionality_wrt_equipollent,
equipollent-identity-right,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-identity-left,
product_functionality_wrt_equipollent_right,
product_functionality_wrt_equipollent_left,
false_wf,
equipollent-unit,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
decidable__le,
decidable__lt,
lelt_wf,
equipollent_functionality_wrt_equipollent2,
unit_wf2,
top-equipollent-unit
Rules used in proof :
cut,
productElimination,
independent_functionElimination,
hypothesis,
natural_numberEquality,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
lemma_by_obid,
because_Cache,
dependent_set_memberEquality,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
hypothesisEquality,
lambdaFormation,
independent_isectElimination,
baseClosed,
imageMemberEquality,
introduction,
isect_memberFormation,
universeEquality,
productEquality,
extract_by_obid,
addLevel
Latex:
\mforall{}[A:Type]. (A \mtimes{} \mBbbN{}1 \msim{} A \mwedge{} \mBbbN{}1 \mtimes{} A \msim{} A)
Date html generated:
2019_06_20-PM-02_17_03
Last ObjectModification:
2018_08_24-PM-11_37_00
Theory : equipollence!!cardinality!
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