Nuprl Lemma : mk_oset_wf

[T:Type]. ∀[eq,leq:T ⟶ T ⟶ 𝔹].
  (mk_oset(T;eq;leq) ∈ LOSet) supposing (UniformLinorder(T;a,b.↑(a leq b)) and IsEqFun(T;eq))


Definitions occuring in Statement :  mk_oset: mk_oset(T;eq;leq) loset: LOSet ulinorder: UniformLinorder(T;x,y.R[x; y]) eqfun_p: IsEqFun(T;eq) assert: b bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] infix_ap: y member: t ∈ T function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  mk_oset: mk_oset(T;eq;leq) uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a prop: so_lambda: λ2y.t[x; y] infix_ap: y so_apply: x[s1;s2] ulinorder: UniformLinorder(T;x,y.R[x; y]) and: P ∧ Q uorder: UniformOrder(T;x,y.R[x; y]) loset: LOSet poset: POSet{i} qoset: QOSet dset: DSet poset_sig: PosetSig set_car: |p| pi1: fst(t) set_eq: =b pi2: snd(t) set_leq: a ≤ b set_le: b upreorder: UniformPreorder(T;x,y.R[x; y])
Lemmas referenced :  ulinorder_wf assert_wf eqfun_p_wf bool_wf set_car_wf set_eq_wf upreorder_wf set_leq_wf uanti_sym_wf connex_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut sqequalHypSubstitution hypothesis axiomEquality equalityTransitivity equalitySymmetry extract_by_obid isectElimination thin hypothesisEquality lambdaEquality applyEquality isect_memberEquality because_Cache functionEquality universeEquality productElimination dependent_set_memberEquality dependent_pairEquality productEquality independent_pairFormation setElimination rename

\mforall{}[T:Type].  \mforall{}[eq,leq:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbB{}].
    (mk\_oset(T;eq;leq)  \mmember{}  LOSet)  supposing  (UniformLinorder(T;a,b.\muparrow{}(a  leq  b))  and  IsEqFun(T;eq))

Date html generated: 2018_05_21-PM-03_13_56
Last ObjectModification: 2018_05_19-AM-08_26_35

Theory : sets_1

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