Theorem 6olt9o

Description: Ordinal 6 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

	Ref	Expression
Assertion	6olt9o	⊢ 6o ∈ 9o

Proof

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