Nuprl Lemma : poset-cat-arrow-not-equal
∀I:Cname List. ∀y:nameset(I). ∀c1,c2:cat-ob(poset-cat(I)).
(c1 y ≠ c2 y ⇒ (∀f:cat-arrow(poset-cat(I)) c1 c2. (((c1 y) = 0 ∈ ℕ2) ∧ ((c2 y) = 1 ∈ ℕ2))))
Proof
Definitions occuring in Statement :
poset-cat: poset-cat(J),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
nequal: a ≠ b ∈ T ,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
name-morph: name-morph(I;J),
so_lambda: λ2x.t[x],
so_apply: x[s],
cat-arrow: cat-arrow(C),
pi2: snd(t),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
false: False,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
le: A ≤ B,
not: ¬A
Lemmas referenced :
cat-arrow_wf,
poset-cat_wf,
nequal_wf,
nameset_wf,
extd-nameset_wf,
nil_wf,
coordinate_name_wf,
all_wf,
assert_wf,
isname_wf,
equal_wf,
cat-ob_wf,
list_wf,
assert_of_le_int,
int_seg_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
le_wf,
int_seg_subtype,
false_wf,
int_seg_cases,
intformand_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
applyEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
intEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
functionEquality,
because_Cache,
functionExtensionality,
dependent_functionElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
unionElimination,
instantiate,
cumulativity,
dependent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
promote_hyp,
hypothesis_subsumption,
addEquality,
int_eqEquality
Latex:
\mforall{}I:Cname List. \mforall{}y:nameset(I). \mforall{}c1,c2:cat-ob(poset-cat(I)).
(c1 y \mneq{} c2 y {}\mRightarrow{} (\mforall{}f:cat-arrow(poset-cat(I)) c1 c2. (((c1 y) = 0) \mwedge{} ((c2 y) = 1))))
Date html generated:
2017_10_05-AM-10_27_54
Last ObjectModification:
2017_07_28-AM-11_23_24
Theory : cubical!sets
Home
Index