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Theorem 2bornot2b 20837
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.)
Assertion
Ref Expression
2bornot2b  |-  ( 2  x.  B  \/  -.  2  x.  B )

Proof of Theorem 2bornot2b
StepHypRef Expression
1 ax-1 5 . . 3  |-  ( -.  2  x.  B  -> 
( 2  x.  B  ->  -.  2  x.  B
) )
2 ax-1 5 . . 3  |-  ( -.  2  x.  B  -> 
( ( 2  x.  B  ->  -.  2  x.  B )  ->  -.  2  x.  B )
)
31, 2mpd 14 . 2  |-  ( -.  2  x.  B  ->  -.  2  x.  B
)
4 df-or 359 . 2  |-  ( ( 2  x.  B  \/  -.  2  x.  B
)  <->  ( -.  2  x.  B  ->  -.  2  x.  B ) )
53, 4mpbir 200 1  |-  ( 2  x.  B  \/  -.  2  x.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357   class class class wbr 4023    x. cmul 8742   2c2 9795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359
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