Theorem predonOLD

Description: Obsolete version of predon as of 16-Oct-2024. (Contributed by Scott Fenton, 27-Mar-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	predonOLD	$$⊢A\in \mathrm{On}\to Pred\left(\mathrm{E},\mathrm{On},A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		predep	$$⊢A\in \mathrm{On}\to Pred\left(\mathrm{E},\mathrm{On},A\right)=\mathrm{On}\cap A$$
2		onss	$$⊢A\in \mathrm{On}\to A\subseteq \mathrm{On}$$
3		sseqin2	$$⊢A\subseteq \mathrm{On}\leftrightarrow \mathrm{On}\cap A=A$$
4	2 3	sylib	$$⊢A\in \mathrm{On}\to \mathrm{On}\cap A=A$$
5	1 4	eqtrd	$$⊢A\in \mathrm{On}\to Pred\left(\mathrm{E},\mathrm{On},A\right)=A$$