Theorem ressucdifsn2

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn . (Contributed by Peter Mazsa, 24-Jul-2024)

	Ref	Expression
Assertion	ressucdifsn2	$$⊢\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$

Proof

Step	Hyp	Ref	Expression
1		disjcsn	$$⊢A\cap \left\{A\right\}=\varnothing$$
2		disjresundif	$$⊢A\cap \left\{A\right\}=\varnothing \to \left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$
3	1 2	ax-mp	$$⊢\left({R\upharpoonright }_{\left(A\cup \left\{A\right\}\right)}\right)\setminus \left({R\upharpoonright }_{\left\{A\right\}}\right)={R\upharpoonright }_A$$