Theorem 1olt9o

Description: Ordinal 1 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

	Ref	Expression
Assertion	1olt9o	⊢ 1o ∈ 9o

Proof

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