Theorem zfauscl

Description: Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep , we invoke the Axiom of Extensionality (indirectly via vtocl ), which is needed for the justification of class variable notation.

If we omit the requirement that y not occur in ph , we can derive a contradiction, as notzfaus shows. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	zfauscl.1	$$⊢A\in \mathrm{V}$$
Assertion	zfauscl	$$⊢\exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		zfauscl.1	$$⊢A\in \mathrm{V}$$
2		eleq2	$$⊢z=A\to \left(x\in z\leftrightarrow x\in A\right)$$
3	2	anbi1d	$$⊢z=A\to \left(\left(x\in z\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$
4	3	bibi2d	$$⊢z=A\to \left(\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\leftrightarrow \left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
5	4	albidv	$$⊢z=A\to \left(\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\leftrightarrow \forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
6	5	exbidv	$$⊢z=A\to \left(\exists y\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)\leftrightarrow \exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)\right)$$
7		ax-sep	$$⊢\exists y\forall x\left(x\in y\leftrightarrow \left(x\in z\wedge \phi \right)\right)$$
8	1 6 7	vtocl	$$⊢\exists y\forall x\left(x\in y\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$