Nuprl Lemma : set_lt_is_sp_of_leq
∀[p:PosetSig]. ∀[a,b:|p|]. uiff(a <p b;strict_part(x,y.x ≤ y;a;b))
Proof
Definitions occuring in Statement :
set_lt: a <p b,
set_leq: a ≤ b,
set_car: |p|,
poset_sig: PosetSig,
strict_part: strict_part(x,y.R[x; y];a;b),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
strict_part: strict_part(x,y.R[x; y];a;b),
set_leq: a ≤ b,
infix_ap: x f y,
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
set_lt: a <p b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
set_blt: a <b b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
assert_witness,
set_le_wf,
set_leq_wf,
set_lt_wf,
set_blt_wf,
strict_part_wf,
set_car_wf,
poset_sig_wf,
and_wf,
not_wf,
assert_wf,
band_wf,
bnot_wf,
uiff_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_set_leq,
assert_of_bnot
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,
applyEquality,
hypothesis,
independent_functionElimination,
lambdaEquality,
dependent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
voidElimination,
independent_pairFormation,
lambdaFormation,
cumulativity,
addLevel,
independent_isectElimination,
impliesFunctionality
Latex:
\mforall{}[p:PosetSig]. \mforall{}[a,b:|p|]. uiff(a <p b;strict\_part(x,y.x \mleq{} y;a;b))
Date html generated:
2016_05_15-PM-00_04_22
Last ObjectModification:
2015_12_26-PM-11_28_55
Theory : sets_1
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