Theorem dford5reg

Description: Given ax-reg , an ordinal is a transitive class totally ordered by the membership relation. (Contributed by Scott Fenton, 28-Jan-2011)

	Ref	Expression
Assertion	dford5reg	⊢ ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ E Or 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ord	⊢ ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ E We 𝐴 ) )
2		zfregfr	⊢ E Fr 𝐴
3		df-we	⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ E Or 𝐴 ) )
4	2 3	mpbiran	⊢ ( E We 𝐴 ↔ E Or 𝐴 )
5	4	anbi2i	⊢ ( ( Tr 𝐴 ∧ E We 𝐴 ) ↔ ( Tr 𝐴 ∧ E Or 𝐴 ) )
6	1 5	bitri	⊢ ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ E Or 𝐴 ) )