How Can Predicate Calculus Prove a Unique Solution in Set Theory?

[poutsos.A]

How do we prove in predicate calculus using the laws of universal end existential quantifiers,propositional calculus,and those of algebra the following??

There exists a unique x, xε{ 2,4,6} such that if yε{ 0,1,2} then x$$^{2}$$y<10.
or in quantifier form:


$$\exists !x$$[ xεA & $$\forall y$$(yεB------> x$$^{2}$$y<10)]

where A={ 2,4,6} and B={ 0,1,2}

 

[evagelos]

poutsos.A

I will try to give a proof of the above problem without mentioning the laws of logic, theorems or axioms and definitions used,you will have to do that.

By a theorem in predicate calculus (with equality) we have.

$$\forall z\exists !x$$(x=z)....................1

and for z=2 we have

$$\exists x$$(x=2)......................2

drop the existential quantifier and

x=2.....................3

but x=2 ====> x=2 v x=4 v x=6......................4

and from 3 and 4 we have : x=2 v x=4 v x=6............5

but xεA <====> x=2 v x=4 v x=6......................6

and from 5 and 6 we get: xεA......................7

now let yεB............................8

But yεB <====> y=0 v y=1 v y=2.......................9

and from 8 and 9 we get: y=0 v y=1 v y=2.....................10

Now let y=0.....................11

but y=0===> y^2=0====>.x$$^{2}$$y=0<10.....................12

and hence y=0 =====> x$$^{2}$$y<10.....................13

in a similar way we prove .

y=1 ====>x$$^{2}$$y<10................14

y=2 =====>x$$^{2}$$y<10................15

hence: y=0 v y=1 v y=2=======>x$$^{2}$$y<10............16

and from 10 and 16 we get: x$$^{2}$$y<10..............17

hence : yεB======>x$$^{2}$$y<10..............18

And introducing universal quantification: $$\forall y$$( yεB====>x$$^{2}$$y<10)..............19

And thus: xεA & $$\forall y$$( yεB====>x$$^{2}$$y<10).........20

And introducing existential quantification we get; $$\exists x$$[ xεA & $$\forall y$$( yεB====>x$$^{2}$$y<10)]...............21

NOW for the uniqueness part you have to prove that.


$$\forall x\forall w$${[ xεA & $$\forall y$$(yεΒ=====>x$$^{2}$$y<10)] & [ wεA & $$\forall y$$(yεΒ=====>w$$^{2}$$y<10)] =====> x=w}