Nuprl Lemma : equipollent-distinct-representatives
∀[A:Type]. ∀[E:A ⟶ A ⟶ ℙ].
(EquivRel(A;x,y.E[x;y]) ⇒ (∀L:A List. (∀a:A. (∃b∈L. E[a;b])) ⇒ x,y:A//E[x;y] ~ ℕ||L|| supposing (∀a,b∈L. ¬E[a;b])))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
pairwise: (∀x,y∈L. P[x; y]),
l_exists: (∃x∈L. P[x]),
length: ||as||,
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
pairwise: (∀x,y∈L. P[x; y]),
not: ¬A,
false: False,
so_apply: x[s1;s2],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
l_exists: (∃x∈L. P[x]),
ge: i ≥ j ,
nat: ℕ,
pi1: fst(t),
le: A ≤ B,
equiv_rel: EquivRel(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
trans: Trans(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
equipollent: A ~ B,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
surject: Surj(A;B;f),
true: True,
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
all_wf,
l_exists_wf,
l_member_wf,
pairwise_wf2,
not_wf,
list_wf,
equiv_rel_wf,
exists_wf,
non_neg_length,
length_wf_nat,
nat_properties,
equal_wf,
lelt_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_wf,
squash_wf,
le_wf,
less_than_wf,
and_wf,
equal-wf-base,
quotient_wf,
biject_wf,
quotient-member-eq,
subtype_quotient,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
applyEquality,
functionExtensionality,
cumulativity,
extract_by_obid,
isectElimination,
because_Cache,
setElimination,
rename,
hypothesis,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
setEquality,
instantiate,
functionEquality,
universeEquality,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination,
dependent_set_memberEquality,
productEquality,
hyp_replacement,
imageMemberEquality,
baseClosed,
pointwiseFunctionalityForEquality,
pertypeElimination,
pointwiseFunctionality
Latex:
\mforall{}[A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(EquivRel(A;x,y.E[x;y])
{}\mRightarrow{} (\mforall{}L:A List. (\mforall{}a:A. (\mexists{}b\mmember{}L. E[a;b])) {}\mRightarrow{} x,y:A//E[x;y] \msim{} \mBbbN{}||L|| supposing (\mforall{}a,b\mmember{}L. \mneg{}E[a;b])))
Date html generated:
2017_04_17-AM-09_33_01
Last ObjectModification:
2017_02_27-PM-05_33_19
Theory : equipollence!!cardinality!
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