Theorem 2olt9o

Description: Ordinal 2 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

	Ref	Expression
Assertion	2olt9o	⊢ 2o ∈ 9o

Proof

Step	Hyp	Ref	Expression
1		2on	⊢ 2o ∈ On
2		sucidg	⊢ ( 2o ∈ On → 2o ∈ suc 2o )
3	1 2	ax-mp	⊢ 2o ∈ suc 2o
4		df-3o	⊢ 3o = suc 2o
5	3 4	eleqtrri	⊢ 2o ∈ 3o
6		3olt9o	⊢ 3o ∈ 9o
7		9on	⊢ 9o ∈ On
8		ontr1	⊢ ( 9o ∈ On → ( ( 2o ∈ 3o ∧ 3o ∈ 9o ) → 2o ∈ 9o ) )
9	7 8	ax-mp	⊢ ( ( 2o ∈ 3o ∧ 3o ∈ 9o ) → 2o ∈ 9o )
10	5 6 9	mp2an	⊢ 2o ∈ 9o