Theorem predon

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

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

Proof

Step	Hyp	Ref	Expression
1		predep	⊢ ( 𝐴 ∈ On → Pred ( E , On , 𝐴 ) = ( On ∩ 𝐴 ) )
2		onss	⊢ ( 𝐴 ∈ On → 𝐴 ⊆ On )
3		sseqin2	⊢ ( 𝐴 ⊆ On ↔ ( On ∩ 𝐴 ) = 𝐴 )
4	2 3	sylib	⊢ ( 𝐴 ∈ On → ( On ∩ 𝐴 ) = 𝐴 )
5	1 4	eqtrd	⊢ ( 𝐴 ∈ On → Pred ( E , On , 𝐴 ) = 𝐴 )