Theorem predon

Description: The predecessor of an ordinal under _E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011) (Proof shortened by BJ, 16-Oct-2024)

	Ref	Expression
Assertion	predon	⊢ ( 𝐴 ∈ On → Pred ( E , On , 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		tron	⊢ Tr On
2		trpred	⊢ ( ( Tr On ∧ 𝐴 ∈ On ) → Pred ( E , On , 𝐴 ) = 𝐴 )
3	1 2	mpan	⊢ ( 𝐴 ∈ On → Pred ( E , On , 𝐴 ) = 𝐴 )