Theorem ontr2

Description: Transitive law for ordinal numbers. Exercise 3 of TakeutiZaring p. 40. (Contributed by NM, 6-Nov-2003)

	Ref	Expression
Assertion	ontr2	⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
2		eloni	⊢ ( 𝐶 ∈ On → Ord 𝐶 )
3		ordtr2	⊢ ( ( Ord 𝐴 ∧ Ord 𝐶 ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )