Nuprl Lemma : set_leq_iff_lt_or_eq
∀s:POSet{i}. ∀a,b:|s|. (a ≤ b ⇐⇒ (a <s b) ∨ (a = b ∈ |s|))
Proof
Definitions occuring in Statement :
poset: POSet{i},
set_lt: a <p b,
set_leq: a ≤ b,
set_car: |p|,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
poset: POSet{i},
qoset: QOSet,
dset: DSet,
ab_binrel: x,y:T. E[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
set_lt: a <p b,
set_blt: a <b b,
infix_ap: x f y,
cand: A c∧ B,
not: ¬A,
guard: {T},
uimplies: b supposing a,
false: False,
uiff: uiff(P;Q)
Lemmas referenced :
set_car_wf,
poset_wf,
poset_properties,
qoset_wf,
uanti_sym_wf,
set_leq_wf,
qoset_properties,
dset_wf,
upreorder_wf,
dset_properties,
poset_sig_wf,
eqfun_p_wf,
set_eq_wf,
or_wf,
set_lt_wf,
equal_wf,
decidable__dset_eq,
assert_wf,
band_wf,
set_le_wf,
bnot_wf,
and_wf,
not_wf,
set_leq_antisymmetry,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_set_leq,
assert_of_bnot,
set_leq_weakening_lt,
set_leq_weakening_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
applyEquality,
lambdaEquality,
setEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
inrFormation,
inlFormation,
because_Cache,
independent_functionElimination,
independent_isectElimination,
voidElimination,
productElimination,
impliesFunctionality
Latex:
\mforall{}s:POSet\{i\}. \mforall{}a,b:|s|. (a \mleq{} b \mLeftarrow{}{}\mRightarrow{} (a <s b) \mvee{} (a = b))
Date html generated:
2016_05_15-PM-00_05_12
Last ObjectModification:
2015_12_26-PM-11_28_28
Theory : sets_1
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