Nuprl Lemma : equipollent-one-iff
∀[A:Type]. (A ~ ℕ1 ⇐⇒ ∃z:A. ∀x:A. (x = z ∈ A))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
rev_implies: P ⇐ Q,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
subtype_rel: A ⊆r B,
guard: {T},
inject: Inj(A;B;f),
true: True,
squash: ↓T,
less_than: a < b,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
all: ∀x:A. B[x],
surject: Surj(A;B;f),
biject: Bij(A;B;f),
exists: ∃x:A. B[x],
equipollent: A ~ B
Lemmas referenced :
equal-wf-base,
equal-wf-base-T,
biject_wf,
false_wf,
lelt_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equipollent_wf,
int_seg_wf,
exists_wf,
all_wf,
equal_wf
Rules used in proof :
universeEquality,
lambdaEquality,
sqequalRule,
hypothesis,
natural_numberEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
independent_isectElimination,
unionElimination,
because_Cache,
intEquality,
setEquality,
rename,
setElimination,
applyEquality,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
introduction,
dependent_set_memberEquality,
dependent_functionElimination,
productElimination,
cumulativity
Latex:
\mforall{}[A:Type]. (A \msim{} \mBbbN{}1 \mLeftarrow{}{}\mRightarrow{} \mexists{}z:A. \mforall{}x:A. (x = z))
Date html generated:
2018_05_21-PM-00_52_47
Last ObjectModification:
2017_12_07-PM-06_30_04
Theory : equipollence!!cardinality!
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