Theorem 2omomeqom

Description: Ordinal two times omega is omega. Lemma 3.17 of Schloeder p. 10. (Contributed by RP, 30-Jan-2025)

	Ref	Expression
Assertion	2omomeqom	⊢ ( 2o ·o ω ) = ω

Proof

Step	Hyp	Ref	Expression
1		omelon	⊢ ω ∈ On
2		2onn	⊢ 2o ∈ ω
3		0ex	⊢ ∅ ∈ V
4	3	prid1	⊢ ∅ ∈ { ∅ , { ∅ } }
5		df2o2	⊢ 2o = { ∅ , { ∅ } }
6	4 5	eleqtrri	⊢ ∅ ∈ 2o
7		omabslem	⊢ ( ( ω ∈ On ∧ 2o ∈ ω ∧ ∅ ∈ 2o ) → ( 2o ·o ω ) = ω )
8	1 2 6 7	mp3an	⊢ ( 2o ·o ω ) = ω