Theorem wesn

Description: Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014)

	Ref	Expression
Assertion	wesn	$$⊢\mathrm{Rel}R\to \left(R\mathrm{We}\left\{A\right\}\leftrightarrow \neg ARA\right)$$

Proof

Step	Hyp	Ref	Expression
1		frsn	$$⊢\mathrm{Rel}R\to \left(R\mathrm{Fr}\left\{A\right\}\leftrightarrow \neg ARA\right)$$
2		sosn	$$⊢\mathrm{Rel}R\to \left(R\mathrm{Or}\left\{A\right\}\leftrightarrow \neg ARA\right)$$
3	1 2	anbi12d	$$⊢\mathrm{Rel}R\to \left(\left(R\mathrm{Fr}\left\{A\right\}\wedge R\mathrm{Or}\left\{A\right\}\right)\leftrightarrow \left(\neg ARA\wedge \neg ARA\right)\right)$$
4		df-we	$$⊢R\mathrm{We}\left\{A\right\}\leftrightarrow \left(R\mathrm{Fr}\left\{A\right\}\wedge R\mathrm{Or}\left\{A\right\}\right)$$
5		pm4.24	$$⊢\neg ARA\leftrightarrow \left(\neg ARA\wedge \neg ARA\right)$$
6	3 4 5	3bitr4g	$$⊢\mathrm{Rel}R\to \left(R\mathrm{We}\left\{A\right\}\leftrightarrow \neg ARA\right)$$