### Nuprl Lemma : rel_plus_irreflexive

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  (WellFnd{i}(T;x,y.x R y) `` (∀x:T. (¬(x R+ x))))`

Proof

Definitions occuring in Statement :  rel_plus: `R+` wellfounded: `WellFnd{i}(A;x,y.R[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]` not: `¬A` false: `False` wellfounded: `WellFnd{i}(A;x,y.R[x; y])` so_lambda: `λ2x.t[x]` infix_ap: `x f y` prop: `ℙ` subtype_rel: `A ⊆r B` so_apply: `x[s]` guard: `{T}` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` or: `P ∨ Q` exists: `∃x:A. B[x]` and: `P ∧ Q` uimplies: `b supposing a` trans: `Trans(T;x,y.E[x; y])`
Lemmas referenced :  not_wf rel_plus_wf all_wf wellfounded_wf rel_plus_implies wellfounded-irreflexive rel_plus_trans rel-rel-plus
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin sqequalHypSubstitution hypothesis isectElimination sqequalRule lambdaEquality lemma_by_obid applyEquality hypothesisEquality because_Cache independent_functionElimination voidElimination functionEquality dependent_functionElimination cumulativity universeEquality isect_memberEquality unionElimination productElimination independent_isectElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (WellFnd\{i\}(T;x,y.x  R  y)  {}\mRightarrow{}  (\mforall{}x:T.  (\mneg{}(x  R\msupplus{}  x))))

Date html generated: 2016_05_14-PM-03_53_55
Last ObjectModification: 2015_12_26-PM-06_56_27

Theory : relations2

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