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	$$⊢\phi \to A\in \mathrm{On}$$
	ontr2d.2	$$⊢\phi \to C\in \mathrm{On}$$
	ontr2d.3	$$⊢\phi \to A\subseteq B$$
	ontr2d.4	$$⊢\phi \to B\in C$$
Assertion	ontr2d	$$⊢\phi \to A\in C$$

Proof

Step	Hyp	Ref	Expression
1		ontr2d.1	$$⊢\phi \to A\in \mathrm{On}$$
2		ontr2d.2	$$⊢\phi \to C\in \mathrm{On}$$
3		ontr2d.3	$$⊢\phi \to A\subseteq B$$
4		ontr2d.4	$$⊢\phi \to B\in C$$
5		ontr2	$$⊢\left(A\in \mathrm{On}\wedge C\in \mathrm{On}\right)\to \left(\left(A\subseteq B\wedge B\in C\right)\to A\in C\right)$$
6	1 2 5	syl2anc	$$⊢\phi \to \left(\left(A\subseteq B\wedge B\in C\right)\to A\in C\right)$$
7	3 4 6	mp2and	$$⊢\phi \to A\in C$$