Nuprl Lemma : mset_mem_char
∀s:DSet. ∀x:|s|. ∀a:MSet{s}. x ∈b a = ∃b{s} y ∈ a. (y (=b) x)
Proof
Definitions occuring in Statement :
mset_for: mset_for,
mset_mem: mset_mem,
mset: MSet{s},
bool: 𝔹,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
bor_mon: <𝔹,∨b>,
dset: DSet,
set_eq: =b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
mset: MSet{s},
member: t ∈ T,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
implies: P ⇒ Q,
mset_for: mset_for,
mset_mem: mset_mem,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
dset: DSet,
so_lambda: λ2x.t[x],
infix_ap: x f y,
so_apply: x[s],
grp_car: |g|,
pi1: fst(t),
bor_mon: <𝔹,∨b>,
bool: 𝔹,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
mem: a ∈b as
Lemmas referenced :
bool_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mem_functionality_wrt_permr,
mon_for_wf,
bor_mon_wf,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
subtype_rel_transitivity,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
set_car_wf,
set_eq_wf,
subtype_rel_self,
iff_weakening_equal,
mem_wf,
permr_wf,
list_wf,
mset_wf,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
promote_hyp,
thin,
productElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
rename,
applyEquality,
lambdaEquality_alt,
imageElimination,
isectElimination,
hypothesisEquality,
universeIsType,
instantiate,
universeEquality,
dependent_functionElimination,
because_Cache,
independent_functionElimination,
independent_isectElimination,
setElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
productIsType,
sqequalBase
Latex:
\mforall{}s:DSet. \mforall{}x:|s|. \mforall{}a:MSet\{s\}. x \mmember{}\msubb{} a = \mexists{}\msubb{}\{s\} y \mmember{} a. (y (=\msubb{}) x)
Date html generated:
2020_05_20-AM-09_35_41
Last ObjectModification:
2020_01_08-PM-06_00_17
Theory : mset
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