Theorem ontr2d

Description: Transitive law for ordinal numbers. Exercise 3 of TakeutiZaring p. 40. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026)

	Ref	Expression
Hypotheses	ontr2d.1	⊢ ( 𝜑 → 𝐴 ∈ On )
	ontr2d.2	⊢ ( 𝜑 → 𝐶 ∈ On )
	ontr2d.3	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	ontr2d.4	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
Assertion	ontr2d	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ontr2d.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		ontr2d.2	⊢ ( 𝜑 → 𝐶 ∈ On )
3		ontr2d.3	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
4		ontr2d.4	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
5		ontr2	⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )
6	1 2 5	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )
7	3 4 6	mp2and	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )