Theorem sucdifsn2

Description: Absorption of union with a singleton by difference. (Contributed by Peter Mazsa, 24-Jul-2024)

	Ref	Expression
Assertion	sucdifsn2	$$⊢\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		disjcsn	$$⊢A\cap \left\{A\right\}=\varnothing$$
2		undif5	$$⊢A\cap \left\{A\right\}=\varnothing \to \left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}=A$$
3	1 2	ax-mp	$$⊢\left(A\cup \left\{A\right\}\right)\setminus \left\{A\right\}=A$$