Nuprl Lemma : set_lt_transitivity_1
∀[s:QOSet]. ∀[a,b,c:|s|]. (a <s c) supposing ((b <s c) and (a ≤ b))
Proof
Definitions occuring in Statement :
qoset: QOSet,
set_lt: a <p b,
set_leq: a ≤ b,
set_car: |p|,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
qoset: QOSet,
dset: DSet,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
set_lt: a <p b,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
set_leq: a ≤ b,
infix_ap: x f y,
guard: {T}
Lemmas referenced :
set_lt_is_sp_of_leq,
assert_witness,
set_blt_wf,
set_lt_wf,
set_leq_wf,
set_car_wf,
qoset_wf,
utrans_imp_sp_utrans_a,
set_le_wf,
set_leq_trans
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
applyEquality
Latex:
\mforall{}[s:QOSet]. \mforall{}[a,b,c:|s|]. (a <s c) supposing ((b <s c) and (a \mleq{} b))
Date html generated:
2016_05_15-PM-00_04_54
Last ObjectModification:
2015_12_26-PM-11_28_03
Theory : sets_1
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