Nuprl Lemma : finite-union
∀S,T:Type. (finite(S) ∧ finite(T) ⇐⇒ finite(S + T))
Proof
Definitions occuring in Statement :
finite: finite(T),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
union: left + right,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
finite: finite(T),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
outl: outl(x),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
mapfilter: mapfilter(f;P;L),
no_repeats: no_repeats(T;l),
subtype_rel: A ⊆r B,
guard: {T},
inject: Inj(A;B;f),
btrue: tt,
true: True,
outr: outr(x),
isr: isr(x)
Lemmas referenced :
finite_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
equipollent_wf,
int_seg_wf,
equipollent_functionality_wrt_equipollent2,
equipollent_inversion,
equipollent-add,
union_functionality_wrt_equipollent,
equipollent_weakening_ext-eq,
ext-eq_weakening,
equipollent-iff-list,
mapfilter_wf,
isl_wf,
assert_wf,
length_wf_nat,
length_wf,
no_repeats_wf,
equal_wf,
all_wf,
l_member_wf,
no_repeats_map,
filter_type,
set_wf,
no_repeats_filter,
equal_functionality_wrt_subtype_rel2,
select_wf,
not_wf,
nat_wf,
less_than_wf,
top_wf,
filter_wf5,
subtype_rel_list,
subtype_rel_union,
member-mapfilter,
assert_elim,
bfalse_wf,
btrue_neq_bfalse,
outl_wf,
isr_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
unionEquality,
universeEquality,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
because_Cache,
independent_functionElimination,
setEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberFormation,
applyEquality,
inlEquality,
addLevel,
levelHypothesis,
inrEquality
Latex:
\mforall{}S,T:Type. (finite(S) \mwedge{} finite(T) \mLeftarrow{}{}\mRightarrow{} finite(S + T))
Date html generated:
2017_04_17-AM-09_34_03
Last ObjectModification:
2017_02_27-PM-05_34_31
Theory : equipollence!!cardinality!
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