Nuprl Lemma : is-above-singleton-subtype
∀[A:Type]. ∀[a:A]. ∀[B:Type].  ∀[z:Base]. (is-above(A;a;z) 
⇒ is-above(B;a;z)) supposing {x:A| x = a ∈ A}  ⊆r B
Proof
Definitions occuring in Statement : 
is-above: is-above(T;a;z)
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
set: {x:A| B[x]} 
, 
base: Base
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
is-above: is-above(T;a;z)
, 
exists: ∃x:A. B[x]
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
squash: ↓T
Lemmas referenced : 
is-above-subtype, 
equal_wf, 
is-above_wf, 
istype-base, 
subtype_rel_wf, 
istype-universe, 
istype-sqle
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :isect_memberFormation_alt, 
cut, 
introduction, 
sqequalRule, 
axiomEquality, 
hypothesis, 
thin, 
rename, 
Error :lambdaFormation_alt, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
setEquality, 
hypothesisEquality, 
independent_isectElimination, 
Error :dependent_set_memberEquality_alt, 
Error :equalityIstype, 
Error :inhabitedIsType, 
independent_functionElimination, 
Error :universeIsType, 
instantiate, 
universeEquality, 
productElimination, 
Error :dependent_pairFormation_alt, 
because_Cache, 
independent_pairFormation, 
Error :productIsType, 
Error :setIsType, 
sqequalBase, 
equalitySymmetry, 
applyLambdaEquality, 
setElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination
Latex:
\mforall{}[A:Type].  \mforall{}[a:A].  \mforall{}[B:Type].
    \mforall{}[z:Base].  (is-above(A;a;z)  {}\mRightarrow{}  is-above(B;a;z))  supposing  \{x:A|  x  =  a\}    \msubseteq{}r  B
Date html generated:
2019_06_20-PM-00_28_13
Last ObjectModification:
2019_01_20-PM-02_37_17
Theory : subtype_1
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