Theorem trssord

Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994)

	Ref	Expression
Assertion	trssord	⊢ ( ( Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵 ) → Ord 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		wess	⊢ ( 𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴 ) )
2		ordwe	⊢ ( Ord 𝐵 → E We 𝐵 )
3	1 2	impel	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ Ord 𝐵 ) → E We 𝐴 )
4	3	anim2i	⊢ ( ( Tr 𝐴 ∧ ( 𝐴 ⊆ 𝐵 ∧ Ord 𝐵 ) ) → ( Tr 𝐴 ∧ E We 𝐴 ) )
5	4	3impb	⊢ ( ( Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵 ) → ( Tr 𝐴 ∧ E We 𝐴 ) )
6		df-ord	⊢ ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ E We 𝐴 ) )
7	5 6	sylibr	⊢ ( ( Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵 ) → Ord 𝐴 )