Nuprl Lemma : set_leq_complement
∀[s:LOSet]. ∀[a,b:|s|]. uiff(¬(a ≤ b);b <s a)
Proof
Definitions occuring in Statement :
loset: LOSet,
set_lt: a <p b,
set_leq: a ≤ b,
set_car: |p|,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
set_lt: a <p b,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
implies: P ⇒ Q,
prop: ℙ,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
ulinorder: UniformLinorder(T;x,y.R[x; y]),
uorder: UniformOrder(T;x,y.R[x; y]),
cand: A c∧ B,
set_leq: a ≤ b,
infix_ap: x f y,
upreorder: UniformPreorder(T;x,y.R[x; y])
Lemmas referenced :
assert_witness,
set_blt_wf,
not_wf,
set_leq_wf,
set_lt_wf,
set_car_wf,
loset_wf,
iff_weakening_uiff,
strict_part_wf,
set_lt_is_sp_of_leq,
uiff_wf,
ulinorder_le_neg,
loset_properties,
poset_properties,
qoset_properties,
set_leq_trans,
upreorder_wf,
set_le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
lemma_by_obid,
setElimination,
rename,
hypothesis,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
dependent_functionElimination,
because_Cache,
voidElimination,
addLevel,
independent_pairFormation,
independent_isectElimination,
cumulativity,
dependent_set_memberEquality,
applyEquality
Latex:
\mforall{}[s:LOSet]. \mforall{}[a,b:|s|]. uiff(\mneg{}(a \mleq{} b);b <s a)
Date html generated:
2016_05_15-PM-00_05_34
Last ObjectModification:
2015_12_26-PM-11_28_17
Theory : sets_1
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