Theorem dford5

Description: A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021)

	Ref	Expression
Assertion	dford5	⊢ ( Ord 𝐴 ↔ ( 𝐴 ⊆ On ∧ Tr 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ordsson	⊢ ( Ord 𝐴 → 𝐴 ⊆ On )
2		ordtr	⊢ ( Ord 𝐴 → Tr 𝐴 )
3	1 2	jca	⊢ ( Ord 𝐴 → ( 𝐴 ⊆ On ∧ Tr 𝐴 ) )
4		epweon	⊢ E We On
5		wess	⊢ ( 𝐴 ⊆ On → ( E We On → E We 𝐴 ) )
6	4 5	mpi	⊢ ( 𝐴 ⊆ On → E We 𝐴 )
7		df-ord	⊢ ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ E We 𝐴 ) )
8	7	biimpri	⊢ ( ( Tr 𝐴 ∧ E We 𝐴 ) → Ord 𝐴 )
9	8	ancoms	⊢ ( ( E We 𝐴 ∧ Tr 𝐴 ) → Ord 𝐴 )
10	6 9	sylan	⊢ ( ( 𝐴 ⊆ On ∧ Tr 𝐴 ) → Ord 𝐴 )
11	3 10	impbii	⊢ ( Ord 𝐴 ↔ ( 𝐴 ⊆ On ∧ Tr 𝐴 ) )