Nuprl Lemma : rneq-by-function
∀x,y,a,b:ℝ. ∀f:ℝ ⟶ ℝ. (a ≠ b ⇒ (f[x] = a) ⇒ (f[y] = b) ⇒ (∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ x ≠ y)
Proof
Definitions occuring in Statement :
rneq: x ≠ y,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
not: ¬A,
uall: ∀[x:A]. B[x],
prop: ℙ,
false: False
Lemmas referenced :
real-weak-Markov,
rneq-cases,
rneq_functionality,
req_wf,
istype-void,
rneq_wf,
real_wf,
rneq_irreflexivity,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_isectElimination,
applyEquality,
independent_functionElimination,
because_Cache,
hypothesis,
productElimination,
unionElimination,
inlFormation_alt,
universeIsType,
isectElimination,
sqequalRule,
functionIsType,
inrFormation_alt,
inhabitedIsType,
voidElimination
Latex:
\mforall{}x,y,a,b:\mBbbR{}. \mforall{}f:\mBbbR{} {}\mrightarrow{} \mBbbR{}.
(a \mneq{} b {}\mRightarrow{} (f[x] = a) {}\mRightarrow{} (f[y] = b) {}\mRightarrow{} (\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f[x] = f[y]))) {}\mRightarrow{} x \mneq{} y)
Date html generated:
2019_10_29-AM-10_23_31
Last ObjectModification:
2019_04_04-AM-11_02_32
Theory : reals
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