# Disjunction

LEMMA    $$P \vee P \vdash P$$

Proof available

LEMMA    $$P \vdash P \vee P$$

Proof available

THEOREM    Disjunction is idempotent. $$P \vee P \dashv\vdash P$$

Proof available

LEMMA    $$P \vee Q \vdash Q \vee P$$

Proof available

LEMMA    $$Q \vee P \vdash P \vee Q$$

Proof available

THEOREM

Disjunction is commutative.

$$P \vee Q \dashv\vdash Q \vee P$$

Proof available

LEMMA    $$P \vee Q \vee R \vdash P \vee (Q \vee R)$$

Proof available

LEMMA    $$P \vee (Q \vee R) \vdash P \vee Q \vee R$$

Proof available

THEOREM

Disjunction is associative.

$$P \vee Q \vee R \dashv\vdash P \vee (Q \vee R)$$

Proof available

LEMMA    $$P \vee (Q \vee R) \vdash (P \vee Q) \vee (P \vee R)$$

Proof available

LEMMA    $$(P \vee Q) \vee (P \vee R) \vdash P \vee (Q \vee R)$$

Proof available

LEMMA

Disjunction is left self-distributive.

$$P \vee (Q \vee R) \dashv\vdash (P \vee Q) \vee (P \vee R)$$

Proof available

LEMMA    $$(Q \vee R) \vee P \vdash (Q \vee P) \vee (R \vee P)$$

Proof available

LEMMA    $$(Q \vee P) \vee (R \vee P) \vdash (Q \vee R) \vee P$$

Proof available

LEMMA

Disjunction is right self-distributive.

$$(Q \vee R) \vee P \dashv\vdash (Q \vee P) \vee (R \vee P)$$

Proof available

THEOREM

Disjunction is self-distributive.

Proof available

THEOREM

False is a left identity element of disjunction.

$$\bot \vee P \dashv\vdash P$$

Proof available

THEOREM

False is a right identity element of disjunction.

$$P \vee \bot \dashv\vdash P$$

Proof available

THEOREM

True is a left zero element of disjunction.

$$\top \vee P \dashv\vdash \top$$

Proof available

THEOREM

True is a right zero element of disjunction.

$$P \vee \top \dashv\vdash \top$$

Proof available