Theorem 3olt9o

Description: Ordinal 3 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

	Ref	Expression
Assertion	3olt9o	⊢ 3o ∈ 9o

Proof

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