The Trinity

Animal = Dog
Animal = Cat
Animal = Bird
BUT
Dog =/= Cat =/= Bird =/= Dog

You're just playing semantics.

It is not literally true that "Dog = Animal", dog is a subset of animals, it is a component of animals. Remember how I said the only way you get around this is with "roughly equal"? That's what you're doing. Using B, C, and D as subsets of category A means that it is not literally true in any case to say that A = B or A = C or A = D, at best you could say A ~=~ B or that B is of A.

A dog is not an animal?
The absolute state of non-trinitarians
So you're asking for something created which is exactly like God? Would you like me to show you an example of an eternal creature? How about an omnipresent creature? The only way you can believe what you're asking for can even possibly exist is if you're a pagan that believes God(s) is fundamentally creaturely.

My father is a human.
I am a human also.
We share human nature, yet we are two different persons.
Same goes with the trinity:
The Father is Divine
The Son is Divine, yet are two different persons.
The Father, Son and Holy Spirit have the same substance or being but are not the same person.
Like how my father and I share the same "being", that is "human being".

Now explain to me how a Divine person (The Father) can beget a Son other than His own substance. Does not human beget only human? Or cat beget only cat?

Animal is the genus of dog
This is not semantics is literally basic logic.

Non-trinitarians everyone

You're just intentionally seizing upon the ambiguity of the English language. It's true that "a dog is an animal", which more explicitly means that an individual dog is an instance of the category of "animal",
Dog = An instance of an animal, animal being a category of life
Category of Dog =/= Category of Animal
What you're saying is
Dog = Subcategory of Animal
Cat = Subcategory of Animal
Bird = Subcategory of Animal
And Subcategory Dog =/= Subcategory Cat. No fucking shit.

No I'm asking for any instance of an equally convoluted logical scenario. You can't think of one because creating an instance of a logical impossibility is impossible.

Irrelevant to my point :^)

No I'm pointing out that the logical relationship of the trinity is contradictory because of the meaning of the word "is" and the phrase "is not". And I'm pointing out that there is no real world example that exemplifies this logically contradictory relationship than can even be rationally imagined.

And to elaborate, this isn't like the trinity.

The Trinity does not say "The Father is a subset of God, The Son is a subset of God, The Holy Spirit is a subset of God, the subsets are equal in power and magnificence but are distinct entities." That WOULD actually be logically possible.

See

That example with dogs and birds is nothing like the logical relationship of the Trinity.

You failed to respond to the first 3/4ths of my post.

Let me understand your position: are you denying the Divinity of Christ or are you claiming Jesus is the same person as The Father?

Father = subset of category human
You = subset of category human
This is not the same thing as "FATHER = HUMAN, YOU = HUMAN, BUT YOU =/= FATHER". In this context "=" means is. You are only one component of the totality of humanity, hence the category human =/= any individual human.

Having the same substance is different from being logically equivalent. Tritheists such as yourself assert that
God is The Father
God is The Son
God is The Holy Spirit
but The Father is not The Son is not The Holy Spirit is not the Father
God is The Son means that the totality of The Son has to be contained by the category God and vice versa. And for this to be true that would mean The Son is The Father is The Holy Spirit. But tritheists deny that.

I'm arguing that The Trinity in and of itself is logically impossible due to the meaning of "is" and "is not."