Theorem difprsnss

Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Assertion	difprsnss	$$⊢\left\{A,B\right\}\setminus \left\{A\right\}\subseteq \left\{B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2	1	elpr	$$⊢x\in \left\{A,B\right\}\leftrightarrow \left(x=A\vee x=B\right)$$
3		velsn	$$⊢x\in \left\{A\right\}\leftrightarrow x=A$$
4	3	notbii	$$⊢\neg x\in \left\{A\right\}\leftrightarrow \neg x=A$$
5		biorf	$$⊢\neg x=A\to \left(x=B\leftrightarrow \left(x=A\vee x=B\right)\right)$$
6	5	biimparc	$$⊢\left(\left(x=A\vee x=B\right)\wedge \neg x=A\right)\to x=B$$
7	2 4 6	syl2anb	$$⊢\left(x\in \left\{A,B\right\}\wedge \neg x\in \left\{A\right\}\right)\to x=B$$
8		eldif	$$⊢x\in \left(\left\{A,B\right\}\setminus \left\{A\right\}\right)\leftrightarrow \left(x\in \left\{A,B\right\}\wedge \neg x\in \left\{A\right\}\right)$$
9		velsn	$$⊢x\in \left\{B\right\}\leftrightarrow x=B$$
10	7 8 9	3imtr4i	$$⊢x\in \left(\left\{A,B\right\}\setminus \left\{A\right\}\right)\to x\in \left\{B\right\}$$
11	10	ssriv	$$⊢\left\{A,B\right\}\setminus \left\{A\right\}\subseteq \left\{B\right\}$$