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	|- ( ph -> A e. On )
	ontr2d.2	|- ( ph -> C e. On )
	ontr2d.3	|- ( ph -> A C_ B )
	ontr2d.4	|- ( ph -> B e. C )
Assertion	ontr2d	|- ( ph -> A e. C )

Proof

Step	Hyp	Ref	Expression
1		ontr2d.1	|- ( ph -> A e. On )
2		ontr2d.2	|- ( ph -> C e. On )
3		ontr2d.3	|- ( ph -> A C_ B )
4		ontr2d.4	|- ( ph -> B e. C )
5		ontr2	|- ( ( A e. On /\ C e. On ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) )
6	1 2 5	syl2anc	|- ( ph -> ( ( A C_ B /\ B e. C ) -> A e. C ) )
7	3 4 6	mp2and	|- ( ph -> A e. C )