### Nuprl Lemma : axiom-choice-0X-quot

`∀B:{B:Type| B ⊆r Base} . ∀X:Type. ∀P:B ⟶ X ⟶ ℙ.  ((∀n:B. ⇃(∃m:X. (P n m))) `` ⇃(∃f:B ⟶ X. ∀n:B. (P n (f n))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` subtype_rel: `A ⊆r B` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` base: `Base` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` sq_stable: `SqStable(P)` squash: `↓T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` uimplies: `b supposing a` subtype_rel: `A ⊆r B`
Lemmas referenced :  subtype_rel_wf subtype_rel_set equiv_rel_true true_wf exists_wf quotient_wf all_wf base_wf sq_stable__subtype_rel canonicalizable-base canonicalizable_wf trivial-quotient-true axiom-choice-quot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin setElimination rename hypothesisEquality independent_functionElimination isectElimination hypothesis introduction sqequalRule imageMemberEquality baseClosed imageElimination lambdaEquality cumulativity applyEquality because_Cache independent_isectElimination functionEquality instantiate universeEquality setEquality

Latex:
\mforall{}B:\{B:Type|  B  \msubseteq{}r  Base\}  .  \mforall{}X:Type.  \mforall{}P:B  {}\mrightarrow{}  X  {}\mrightarrow{}  \mBbbP{}.
((\mforall{}n:B.  \00D9(\mexists{}m:X.  (P  n  m)))  {}\mRightarrow{}  \00D9(\mexists{}f:B  {}\mrightarrow{}  X.  \mforall{}n:B.  (P  n  (f  n))))

Date html generated: 2016_05_14-PM-09_42_30
Last ObjectModification: 2016_01_15-PM-10_55_33

Theory : continuity

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