Theorem 4olt9o

Description: Ordinal 4 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

	Ref	Expression
Assertion	4olt9o	⊢ 4o ∈ 9o

Proof

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