### 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))`

Proof

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: `b supposing a` uall: `∀[x:A]. B[x]` infix_ap: `x f 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: `b supposing a` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` infix_ap: `x f 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

Latex:
\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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