Theorem 2bornot2b 8724

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.)

Assertion

Ref	Expression
2bornot2b	|- (2 x. B \/ -. 2 x. B)

Proof of Theorem 2bornot2b

Step	Hyp	Ref	Expression
1		ax-1 4	. . 3 |- (-. 2 x. B -> (2 x. B -> -. 2 x. B))
2		ax-1 4	. . 3 |- (-. 2 x. B -> ((2 x. B -> -. 2 x. B) -> -. 2 x. B))
3	1, 2	mpd 26	. 2 |- (-. 2 x. B -> -. 2 x. B)
4		df-or 224	. 2 |- ((2 x. B \/ -. 2 x. B) <-> (-. 2 x. B -> -. 2 x. B))
5	3, 4	mpbir 190	1 |- (2 x. B \/ -. 2 x. B)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   class class class wbr 2609   x. cmul 5211  2c2 5908

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224

Copyright terms: Public domain