Exercise 1, § 3

Tag 03G9, permanent. Cite it as [Bourbaki, Tag 03G9].

Theory of Sets, I, 57

Let $A$ , $B$ , $C$ be relations in a logical theory $\mathscr{T}$ . Show that the following relations are theorems in $\mathscr{T}$ :

A \Rightarrow (B \Rightarrow A),

(A \Rightarrow B) \Rightarrow ((B \Rightarrow C) \Rightarrow (A \Rightarrow C)),

A \Rightarrow ((\text{not } A) \Rightarrow B),

(A \text{ or } B) \Leftrightarrow ((A \Rightarrow B) \Rightarrow B),

(A \Leftrightarrow B) \Leftrightarrow ((A \text{ and } B) \text{ or } ((\text{not } A) \text{ and } (\text{not } B))),

(A \Leftrightarrow B) \Leftrightarrow \text{not } ((\text{not } A) \Leftrightarrow B),

(A \Rightarrow (B \text{ or } (\text{not } C))) \Leftrightarrow ((C \text{ and } A) \Rightarrow B),

(A \Rightarrow (B \text{ or } C)) \Leftrightarrow (B \text{ or } (A \Rightarrow C)),

(A \Rightarrow B) \Rightarrow ((A \Rightarrow C) \Rightarrow (A \Rightarrow (B \text{ and } C))),

(A \Rightarrow C) \Rightarrow ((B \Rightarrow C) \Rightarrow ((A \text{ or } B) \Rightarrow C)),

(A \Rightarrow B) \Rightarrow ((A \text{ and } C) \Rightarrow (B \text{ and } C)),

(A \Rightarrow B) \Rightarrow ((A \text{ or } C) \Rightarrow (B \text{ or } C)).

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