Two Planes have Line in Common/Hilbert's Axioms

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Theorem

Two planes $\alpha, \beta$ have no point in common or a straight line $a$ in common.

Proof

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By Law of Excluded Middle, $\alpha$ and $\beta$ have no point in common, or some point $A$ in common.

In the former case, the theorem holds trivially.

In the latter, by Axiom $\text I, 6$, there is a second point $B$ lying on both $\alpha$ and $\beta$.

By Axiom $I, 1$, there is a straight line $a$ containing both $A$ and $B$.

Then, by Axiom $I, 5$, every point of $a$ lies on both $\alpha$ and $\beta$.

$\blacksquare$

Sources

• 1902: David Hilbert: The Foundations of Geometry (translated by E.J. Townsend): $\text I \S 2$. Group $1$: Axioms of connection