Theorem sucdifsn2

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

	Ref	Expression
Assertion	sucdifsn2	⊢ ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		disjcsn	⊢ ( 𝐴 ∩ { 𝐴 } ) = ∅
2		undif5	⊢ ( ( 𝐴 ∩ { 𝐴 } ) = ∅ → ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = 𝐴 )
3	1 2	ax-mp	⊢ ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = 𝐴