Nuprl Lemma : setTC_functionality_subset
∀b,a:Set{i:l}. ((a ⊆ b) ⇒ (setTC(a) ⊆ setTC(b)))
Proof
Definitions occuring in Statement :
setsubset: (a ⊆ b),
setTC: setTC(a),
Set: Set{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
set-add: a + b,
pi2: snd(t),
set-item: set-item(s;x),
Wsup: Wsup(a;b),
mk-set: f"(T),
setunionfun: ⋃x∈s.f[x],
top: Top,
exists: ∃x:A. B[x],
guard: {T},
or: P ∨ Q,
setTC: Error :setTC,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
set-item_wf,
seteq_wf,
set-dom_wf,
setmem-iff,
seteq-iff,
setmem-mk-set-sq,
setmem-unionfun-implies,
setunionfun_wf,
setmem-set-add,
set-subtype-coSet,
setmem_wf,
setsubset-iff,
mk-set_wf,
setsubset_wf,
Set_wf,
all_wf,
set-induction
Rules used in proof :
spreadEquality,
dependent_pairEquality,
dependent_pairFormation,
inrFormation,
voidEquality,
voidElimination,
isect_memberEquality,
inlFormation,
unionElimination,
because_Cache,
setEquality,
rename,
setElimination,
productElimination,
dependent_functionElimination,
universeEquality,
applyEquality,
lambdaFormation,
independent_functionElimination,
hypothesisEquality,
functionEquality,
cumulativity,
hypothesis,
instantiate,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}b,a:Set\{i:l\}. ((a \msubseteq{} b) {}\mRightarrow{} (setTC(a) \msubseteq{} setTC(b)))
Date html generated:
2018_07_29-AM-10_03_32
Last ObjectModification:
2018_07_11-PM-10_01_00
Theory : constructive!set!theory
Home
Index