Disjunction of Conditional and Converse/Proof 1

Theorem
Given any two statements, one of them implies the other.


 * $\vdash \left({p \implies q}\right) \lor \left({q \implies p}\right)$

That is, given any conditional, either it is true or its converse is.

Proof

 * align="right" | 1 ||
 * align="right" |
 * $p \lor \neg p$
 * LEM
 * (None)
 * (None)


 * align="right" | 3 ||
 * align="right" | 2
 * $q \implies p$
 * Sequent Introduction
 * 2
 * True Statement is Implied by Every Statement
 * True Statement is Implied by Every Statement


 * align="right" | 6 ||
 * align="right" | 5
 * $p \implies q$
 * SI
 * 5
 * False Statement Implies Every Statement
 * False Statement Implies Every Statement


 * align="right" | 8 ||
 * align="right" |
 * $\left({p \implies q}\right) \lor \left({q \implies p}\right)$
 * $\lor \mathcal E$
 * 1, 2-4, 5-7
 * 1, 2-4, 5-7