Theorem partsuc

Description: Property of the partition. (Contributed by Peter Mazsa, 20-Sep-2024)

	Ref	Expression
Assertion	partsuc	$$⊢\left(\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)\right)Part\left(\mathrm{suc}A\setminus \left\{A\right\}\right)\leftrightarrow \left({R\upharpoonright }_A\right)PartA$$

Proof

Step	Hyp	Ref	Expression
1		ressucdifsn	$$⊢\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$
2		sucdifsn	$$⊢\mathrm{suc}A\setminus \left\{A\right\}=A$$
3		parteq12	$$⊢\left(\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A\wedge \mathrm{suc}A\setminus \left\{A\right\}=A\right)\to \left(\left(\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)\right)Part\left(\mathrm{suc}A\setminus \left\{A\right\}\right)\leftrightarrow \left({R\upharpoonright }_A\right)PartA\right)$$
4	1 2 3	mp2an	$$⊢\left(\left({R\upharpoonright }_{\mathrm{suc}A}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)\right)Part\left(\mathrm{suc}A\setminus \left\{A\right\}\right)\leftrightarrow \left({R\upharpoonright }_A\right)PartA$$