Theorem disjsn

Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998) (Proof shortened by Andrew Salmon, 29-Jun-2011) (Proof shortened by Wolf Lammen, 30-Sep-2014)

	Ref	Expression
Assertion	disjsn	$$⊢A\cap \left\{B\right\}=\varnothing \leftrightarrow \neg B\in A$$

Proof

Step	Hyp	Ref	Expression
1		disj1	$$⊢A\cap \left\{B\right\}=\varnothing \leftrightarrow \forall x\left(x\in A\to \neg x\in \left\{B\right\}\right)$$
2		con2b	$$⊢\left(x\in A\to \neg x\in \left\{B\right\}\right)\leftrightarrow \left(x\in \left\{B\right\}\to \neg x\in A\right)$$
3		velsn	$$⊢x\in \left\{B\right\}\leftrightarrow x=B$$
4	3	imbi1i	$$⊢\left(x\in \left\{B\right\}\to \neg x\in A\right)\leftrightarrow \left(x=B\to \neg x\in A\right)$$
5		imnan	$$⊢\left(x=B\to \neg x\in A\right)\leftrightarrow \neg \left(x=B\wedge x\in A\right)$$
6	2 4 5	3bitri	$$⊢\left(x\in A\to \neg x\in \left\{B\right\}\right)\leftrightarrow \neg \left(x=B\wedge x\in A\right)$$
7	6	albii	$$⊢\forall x\left(x\in A\to \neg x\in \left\{B\right\}\right)\leftrightarrow \forall x\neg \left(x=B\wedge x\in A\right)$$
8		alnex	$$⊢\forall x\neg \left(x=B\wedge x\in A\right)\leftrightarrow \neg \exists x\left(x=B\wedge x\in A\right)$$
9		dfclel	$$⊢B\in A\leftrightarrow \exists x\left(x=B\wedge x\in A\right)$$
10	8 9	xchbinxr	$$⊢\forall x\neg \left(x=B\wedge x\in A\right)\leftrightarrow \neg B\in A$$
11	1 7 10	3bitri	$$⊢A\cap \left\{B\right\}=\varnothing \leftrightarrow \neg B\in A$$