Theorem ordin

Description: The intersection of two ordinal classes is ordinal. Proposition 7.9 of TakeutiZaring p. 37. (Contributed by NM, 9-May-1994)

	Ref	Expression
Assertion	ordin	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → Ord ( 𝐴 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ordtr	⊢ ( Ord 𝐴 → Tr 𝐴 )
2		ordtr	⊢ ( Ord 𝐵 → Tr 𝐵 )
3		trin	⊢ ( ( Tr 𝐴 ∧ Tr 𝐵 ) → Tr ( 𝐴 ∩ 𝐵 ) )
4	1 2 3	syl2an	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → Tr ( 𝐴 ∩ 𝐵 ) )
5		inss2	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵
6		trssord	⊢ ( ( Tr ( 𝐴 ∩ 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵 ∧ Ord 𝐵 ) → Ord ( 𝐴 ∩ 𝐵 ) )
7	5 6	mp3an2	⊢ ( ( Tr ( 𝐴 ∩ 𝐵 ) ∧ Ord 𝐵 ) → Ord ( 𝐴 ∩ 𝐵 ) )
8	4 7	sylancom	⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → Ord ( 𝐴 ∩ 𝐵 ) )