Nuprl Lemma : exists_uni_upto_char
∀T:Type. ∀r:T ⟶ T ⟶ ℙ. ∀Q:T ⟶ ℙ. ((∃x:T. Q[x]) ⇒ (∀x,y:T. (Q[x] ⇒ Q[y] ⇒ (x [r] y))) ⇒ (r)∃!x:T. Q[x])
Proof
Definitions occuring in Statement :
exists_uni_upto: exists_uni_upto,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
binrel_ap: a [r] b
Definitions unfolded in proof :
exists_uni_upto: exists_uni_upto,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
uni_sat_upto: uni_sat_upto,
and: P ∧ Q
Lemmas referenced :
all_wf,
binrel_ap_wf,
exists_wf,
uni_sat_upto_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
functionEquality,
applyEquality,
hypothesis,
cumulativity,
universeEquality,
dependent_pairFormation,
independent_pairFormation,
dependent_functionElimination,
because_Cache,
independent_functionElimination
Latex:
\mforall{}T:Type. \mforall{}r:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}Q:T {}\mrightarrow{} \mBbbP{}.
((\mexists{}x:T. Q[x]) {}\mRightarrow{} (\mforall{}x,y:T. (Q[x] {}\mRightarrow{} Q[y] {}\mRightarrow{} (x [r] y))) {}\mRightarrow{} (r)\mexists{}!x:T. Q[x])
Date html generated:
2016_05_16-AM-07_45_12
Last ObjectModification:
2015_12_28-PM-05_53_37
Theory : factor_1
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