Nuprl Lemma : type-with-x=0-y=1
∀x,y:Base. ∃T:Type. ((x = 0 ∈ T) ∧ (y = 1 ∈ T) ∧ ((0 = 1 ∈ T) ⇒ (↓(x = y ∈ Base) ∨ (x = 1 ∈ Base) ∨ (y = 0 ∈ Base))))
Proof
Definitions occuring in Statement :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n,
base: Base,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
true: True,
false: False,
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
quotient: x,y:A//B[x; y]
Lemmas referenced :
base_wf,
or_wf,
equal-wf-base,
set_wf,
subtype_base_sq,
subtype_rel_self,
int_subtype_base,
quotient_wf,
quotient-member-eq,
squash_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
inrFormation,
inlFormation,
setElimination,
thin,
rename,
hypothesisEquality,
sqequalHypSubstitution,
isectElimination,
productEquality,
because_Cache,
baseClosed,
lambdaEquality,
independent_pairFormation,
unionElimination,
equalitySymmetry,
equalityTransitivity,
productElimination,
instantiate,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
intEquality,
voidElimination,
promote_hyp,
dependent_pairFormation,
setEquality,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
functionEquality,
sqequalIntensionalEquality,
pertypeElimination
Latex:
\mforall{}x,y:Base. \mexists{}T:Type. ((x = 0) \mwedge{} (y = 1) \mwedge{} ((0 = 1) {}\mRightarrow{} (\mdownarrow{}(x = y) \mvee{} (x = 1) \mvee{} (y = 0))))
Date html generated:
2018_05_21-PM-01_14_27
Last ObjectModification:
2018_05_01-PM-04_37_27
Theory : num_thy_1
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