Nuprl Lemma : less_than_transitivity1
∀[x,y,z:ℤ]. (x < z) supposing ((y ≤ z) and x < y)
Proof
Definitions occuring in Statement :
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
or: P ∨ Q,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
le-iff-less-or-equal,
le_wf,
member-less_than,
less_than_wf,
less_than_transitivity,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
unionElimination,
isectElimination,
sqequalRule,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
intEquality,
applyEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_functionElimination
Latex:
\mforall{}[x,y,z:\mBbbZ{}]. (x < z) supposing ((y \mleq{} z) and x < y)
Date html generated:
2019_06_20-AM-11_22_48
Last ObjectModification:
2018_09_10-PM-01_14_11
Theory : arithmetic
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