Theorem 2bornot2b 28729

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 · 𝐵 ∨ ¬ 2 · 𝐵)

Proof of Theorem 2bornot2b

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (¬ 2 · 𝐵 → (2 · 𝐵 → ¬ 2 · 𝐵))
2		ax-1 6	. . 3 ⊢ (¬ 2 · 𝐵 → ((2 · 𝐵 → ¬ 2 · 𝐵) → ¬ 2 · 𝐵))
3	1, 2	mpd 15	. 2 ⊢ (¬ 2 · 𝐵 → ¬ 2 · 𝐵)
4		df-or 844	. 2 ⊢ ((2 · 𝐵 ∨ ¬ 2 · 𝐵) ↔ (¬ 2 · 𝐵 → ¬ 2 · 𝐵))
5	3, 4	mpbir 230	1 ⊢ (2 · 𝐵 ∨ ¬ 2 · 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843   class class class wbr 5070   · cmul 10807  2c2 11958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 844

This theorem is referenced by: (None)