Nuprl Lemma : cosetTC-contains
∀a:coSet{i:l}. (a ⊆ cosetTC(a))
Proof
Definitions occuring in Statement :
setsubset: (a ⊆ b),
cosetTC: cosetTC(a),
coSet: coSet{i:l},
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
cosetTC: cosetTC(a),
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
setmem: (x ∈ s),
coWmem: coWmem(a.B[a];z;w),
exists: ∃x:A. B[x],
seteq: seteq(s1;s2),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
copath-length: copath-length(p),
pi1: fst(t),
copath-cons: copath-cons(b;x),
copath-nil: (),
true: True,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
coSet: coSet{i:l},
subtype_rel: A ⊆r B,
nat: ℕ,
copath-at: copath-at(w;p),
coPath-at: coPath-at(n;w;p),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bfalse: ff,
subtract: n - m,
btrue: tt,
coW-item: coW-item(w;b),
pi2: snd(t),
prop: ℙ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
setmem-mk-coset,
istype-void,
copath-cons_wf,
istype-universe,
copath-nil_wf,
coW-item_wf,
istype-less_than,
copath-length_wf,
coW-equiv_wf,
copath-at_wf,
setmem_wf,
coSet_wf,
setsubset-iff,
cosetTC_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
sqequalRule,
productElimination,
dependent_pairFormation_alt,
independent_pairFormation,
natural_numberEquality,
imageMemberEquality,
hypothesisEquality,
baseClosed,
dependent_set_memberEquality_alt,
universeEquality,
lambdaEquality_alt,
instantiate,
because_Cache,
applyEquality,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
universeIsType,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}a:coSet\{i:l\}. (a \msubseteq{} cosetTC(a))
Date html generated:
2019_10_31-AM-06_33_50
Last ObjectModification:
2018_12_13-PM-02_29_30
Theory : constructive!set!theory
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