Nuprl Lemma : equipollent-subtract2
∀a,b:ℕ. ∀[T:Type]. (T ~ ℕa ⇒ (∀[P:T ⟶ ℙ]. ({x:T| P[x]} ~ ℕb ⇒ {x:T| ¬P[x]} ~ ℕa - b)))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
equipollent: A ~ B,
exists: ∃x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
surject: Surj(A;B;f),
so_lambda: λ2x.t[x]
Lemmas referenced :
equipollent_inversion,
int_seg_wf,
equipollent_wf,
subtype_rel_self,
istype-universe,
istype-nat,
equipollent_functionality_wrt_equipollent2,
biject_wf,
iff_weakening_equal,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
istype-le,
istype-less_than,
equal_functionality_wrt_subtype_rel2,
equipollent-subtract,
not_wf,
subtract_wf,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
productElimination,
universeIsType,
setEquality,
applyEquality,
sqequalRule,
instantiate,
universeEquality,
functionIsType,
inhabitedIsType,
dependent_pairFormation_alt,
functionExtensionality_alt,
dependent_set_memberEquality_alt,
setIsType,
independent_pairFormation,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
equalityIstype,
imageElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
int_eqEquality,
Error :memTop,
voidElimination,
productIsType,
applyLambdaEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}a,b:\mBbbN{}. \mforall{}[T:Type]. (T \msim{} \mBbbN{}a {}\mRightarrow{} (\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. (\{x:T| P[x]\} \msim{} \mBbbN{}b {}\mRightarrow{} \{x:T| \mneg{}P[x]\} \msim{} \mBbbN{}a - b)))
Date html generated:
2020_05_19-PM-10_00_37
Last ObjectModification:
2020_01_04-PM-08_00_22
Theory : equipollence!!cardinality!
Home
Index