Theorem 7olt9o

Description: Ordinal 7 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

	Ref	Expression
Assertion	7olt9o	⊢ 7o ∈ 9o

Proof

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