Nuprl Lemma : poset-cat-ob-cases
∀I:Cname List. ∀c1,c2:cat-ob(poset-cat(I)). ((c1 = c2 ∈ cat-ob(poset-cat(I))) ∨ (∃y:nameset(I). c1 y ≠ c2 y))
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
nameset: nameset(L),
coordinate_name: Cname,
cat-ob: cat-ob(C),
list: T List,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nequal: a ≠ b ∈ T ,
or: P ∨ Q,
apply: f a,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
poset-cat: poset-cat(J),
cat-ob: cat-ob(C),
pi1: fst(t),
nameset: nameset(L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
prop: ℙ,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
cand: A c∧ B,
coordinate_name: Cname,
int_upper: {i...},
nat: ℕ,
le: A ≤ B,
sq_stable: SqStable(P)
Lemmas referenced :
list-subtype,
coordinate_name_wf,
cat-ob_wf,
poset-cat_wf,
list_wf,
nameset_wf,
nequal_wf,
extd-nameset_subtype_int,
nil_wf,
equal_wf,
decidable__l_exists_better-extract,
decidable__not,
decidable__equal_int,
name-morph_wf,
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,
all_wf,
assert_wf,
isname_wf,
extd-nameset_wf,
exists_wf,
subtype_base_sq,
int_seg_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
nsub2_subtype_extd-nameset,
decidable__equal_int_seg,
extd-nameset-nil,
l_exists_iff,
l_member_wf,
le_wf,
select_member,
sq_stable__l_member,
decidable__equal-coordinate_name,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
intEquality,
applyEquality,
setElimination,
rename,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
inrFormation,
productElimination,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
inlFormation,
dependent_set_memberEquality,
functionEquality,
functionExtensionality,
instantiate,
cumulativity,
setEquality,
imageMemberEquality,
baseClosed,
productEquality
Latex:
\mforall{}I:Cname List. \mforall{}c1,c2:cat-ob(poset-cat(I)). ((c1 = c2) \mvee{} (\mexists{}y:nameset(I). c1 y \mneq{} c2 y))
Date html generated:
2017_10_05-AM-10_28_03
Last ObjectModification:
2017_07_28-AM-11_23_28
Theory : cubical!sets
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