Theorem velcomp

Description: Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011)

	Ref	Expression
Assertion	velcomp	$$⊢x\in \left(\mathrm{V}\setminus A\right)\leftrightarrow \neg x\in A$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		eldif	$$⊢x\in \left(\mathrm{V}\setminus A\right)\leftrightarrow \left(x\in \mathrm{V}\wedge \neg x\in A\right)$$
3	1 2	mpbiran	$$⊢x\in \left(\mathrm{V}\setminus A\right)\leftrightarrow \neg x\in A$$