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Metamath Proof Explorer

Theorem xpomen

Description: The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of Enderton p. 133. (Contributed by NM, 23-Jul-2004) (Revised by Mario Carneiro, 9-Mar-2013)

		Ref	Expression
	Assertion	xpomen	$⊢ (ω \times ω) \approx ω$

Proof

Step	Hyp	Ref	Expression
1		omelon	$⊢ ω \in On$
2		ssid	$⊢ ω \subseteq ω$
3		infxpen	$⊢ (ω \in On \land ω \subseteq ω) \to (ω \times ω) \approx ω$
4	1 2 3	mp2an	$⊢ (ω \times ω) \approx ω$