Theorem 2bornot2b

Description: The law of excluded middle. Act III, Theorem 1 of Shakespeare,Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	2bornot2b	$$⊢\left(2\times B\vee \neg 2\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-1	$$⊢\neg 2\times B\to \left(2\times B\to \neg 2\times B\right)$$
2		ax-1	$$⊢\neg 2\times B\to \left(\left(2\times B\to \neg 2\times B\right)\to \neg 2\times B\right)$$
3	1 2	mpd	$$⊢\neg 2\times B\to \neg 2\times B$$
4		df-or	$$⊢\left(2\times B\vee \neg 2\times B\right)\leftrightarrow \left(\neg 2\times B\to \neg 2\times B\right)$$
5	3 4	mpbir	$$⊢\left(2\times B\vee \neg 2\times B\right)$$