Theorem partsuc2

Description: Property of the partition. (Contributed by Peter Mazsa, 24-Jul-2024)

	Ref	Expression
Assertion	partsuc2	$$⊢\left(\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)\right)Part\left(\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}\right)\leftrightarrow \left({R\upharpoonright }_A\right)PartA$$

Proof

Step	Hyp	Ref	Expression
1		ressucdifsn2	$$⊢\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$
2		sucdifsn2	$$⊢\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}=A$$
3		parteq12	$$⊢\left(\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A\wedge \left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}=A\right)\to \left(\left(\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)\right)Part\left(\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}\right)\leftrightarrow \left({R\upharpoonright }_A\right)PartA\right)$$
4	1 2 3	mp2an	$$⊢\left(\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)\right)Part\left(\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}\right)\leftrightarrow \left({R\upharpoonright }_A\right)PartA$$