Theorem 5olt9o

Description: Ordinal 5 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

	Ref	Expression
Assertion	5olt9o	⊢ 5o ∈ 9o

Proof

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