Nuprl Lemma : equipollent-subtract
∀a,b:ℕ. ∀[P:ℕa ⟶ ℙ]. ({x:ℕa| P[x]} ~ ℕb ⇒ {x:ℕa| ¬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
Definitions unfolded in proof :
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
int_seg: {i..j-},
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
equipollent: A ~ B,
or: P ∨ Q,
decidable: Dec(P),
squash: ↓T,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
bfalse: ff,
true: True,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
false: False,
not: ¬A,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
lelt: i ≤ j < k,
ge: i ≥ j ,
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
surject: Surj(A;B;f),
biject: Bij(A;B;f),
inject: Inj(A;B;f)
Lemmas referenced :
subtype_rel_sets,
equal_functionality_wrt_subtype_rel2,
biject_wf,
intformand_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
equipollent_functionality_wrt_equipollent,
equipollent-set,
equipollent_functionality_wrt_equipollent2,
equipollent-nsub,
decidable_functionality,
ext-eq_weakening,
equipollent_weakening_ext-eq,
equipollent-partition,
set_wf,
true_wf,
false_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
sq_stable_from_decidable,
decidable__assert,
isl_wf,
exists_wf,
not_wf,
squash_wf,
assert_wf,
assert_witness,
all_wf,
iff_wf,
decidable__exists_int_seg,
equal_wf,
decidable__equal_int,
equipollent_inversion,
int_seg_wf,
equipollent_wf,
nat_wf
Rules used in proof :
cumulativity,
functionEquality,
universeEquality,
lambdaEquality,
independent_functionElimination,
sqequalRule,
because_Cache,
applyEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
setEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
intEquality,
dependent_functionElimination,
instantiate,
productElimination,
independent_pairEquality,
baseClosed,
imageMemberEquality,
imageElimination,
independent_pairFormation,
introduction,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity,
unionElimination,
unionEquality,
voidElimination,
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
independent_isectElimination,
promote_hyp,
dependent_set_memberEquality
Latex:
\mforall{}a,b:\mBbbN{}. \mforall{}[P:\mBbbN{}a {}\mrightarrow{} \mBbbP{}]. (\{x:\mBbbN{}a| P[x]\} \msim{} \mBbbN{}b {}\mRightarrow{} \{x:\mBbbN{}a| \mneg{}P[x]\} \msim{} \mBbbN{}a - b)
Date html generated:
2018_05_21-PM-00_53_29
Last ObjectModification:
2017_12_07-PM-06_29_16
Theory : equipollence!!cardinality!
Home
Index