∀P∃Y[Y={x|P(x)}] (The Axiom Schema of Comprehension)⇔ ∀P∃Y[x∈Y^P(x)]

∃P[P(x)=x∉Y]

P(x)[P(x) = x∉Y] ⇔ x∈Y

__(false, the very same contradiction born by the former derivation)__
∴¬∀P∃Y[x∈Y^P(x)] (true)

¬∀P∃Y[x∈Y^P(x)] ⇔ ¬∀P∃Y[Y={x|P(x)}] (The Axiom Schema of Comprehension)

**Also, the Axiom Schema of Separation seems to be false due to the following**
∀P∃Y∃A[Y={x∈A|P(x)}] (The Axiom Schema of Separation) ⇔∀P∃Y∃A[x∈A^x∈Y^P(x)]

∃P[P(x)=x∉Y]

P(x)[P(x)= x∉Y] ⇔ x∈Y

__(false, again, the same contradiction)__
∴¬∀P∃Y∃A[x∈A^x∈Y^P(x)] (true)

¬∀P∃Y∃A[x∈A^x∈Y^P(x)] ⇔¬∀P∃Y∃A[Y={x∈A|P(x)}] (The Axiom Schema of Separation)