## Two Empty Sets

 Quote by Erland Your argument shows that I was right, but perhaps that is what you meant? To be clear, you should have written the two last lines as ##\exists x[ (x\in \emptyset\wedge \neg x\in \emptyset')]\vee[(x\in \emptyset'\wedge \neg x\in \emptyset )]## and ##\exists x[ (x\in \emptyset\vee x\in \emptyset')]##. Then, the argument is correct, since ##\exists x(P(x)\vee Q(x))## and ##\exists x \,P(x)\vee\exists x\,Q(x)## are logically equivalent.
Can you support that ,by writing a complete formal proof??

Because i know that it will be useless to ask you ,where did you get the:

"and ##\exists x[ (x\in \emptyset\vee x\in \emptyset')]##" ,part, e.t.c ,e.t.c

 Quote by Erland ##\exists x[ (x\in \emptyset\wedge \neg x\in \emptyset')]\vee[(x\in \emptyset'\wedge \neg x\in \emptyset )]##
Oops, the parenteses here are partially wrong. It should be:

##\exists x[ (x\in \emptyset\wedge \neg x\in \emptyset')\vee(x\in \emptyset'\wedge \neg x\in \emptyset )]##.

 Quote by stauros Can you support that ,by writing a complete formal proof?? Because i know that it will be useless to ask you ,where did you get the: "and ##\exists x[ (x\in \emptyset\vee x\in \emptyset')]##" ,part, e.t.c ,e.t.c
Well, how did you get the corresponding line, in your previous post?

If you want a complete formal proof, you must specify which formal system that should be used: which are the axioms and the rules of inference? Is a Hilbert style axiom system (and which variant in this case) or a natural deduction system (and which variant in this case) or some other kind of system?

And whatever system is used, complete formal proofs tend to be extremely lengthy. One almost always takes shortcuts. But you have a habit of questioning all possible shortcuts.

 Quote by Erland Well, how did you get the corresponding line, in your previous post? If you want a complete formal proof, you must specify which formal system that should be used: which are the axioms and the rules of inference? Is a Hilbert style axiom system (and which variant in this case) or a natural deduction system (and which variant in this case) or some other kind of system? And whatever system is used, complete formal proofs tend to be extremely lengthy. One almost always takes shortcuts. But you have a habit of questioning all possible shortcuts.
Write a formal proof supporting your argument in any system you like using any rules of inference you like ,i can follow.

There is no other way of checking whether your argument is right or wrong.

But before anything else let us give the definition of a formal proof.

A formal proof is : a finite No of statements ,where each statement is either an axiom or a theorem or a definition or a conclusion by appling a rule of inference on one or more previous statements.

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