Theorem eldifeldifsn

Description: An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022)

	Ref	Expression
Assertion	eldifeldifsn	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝑌 ∈ ( 𝐵 ∖ { 𝑋 } ) )

Proof

Step	Hyp	Ref	Expression
1		snssi	⊢ ( 𝑋 ∈ 𝐴 → { 𝑋 } ⊆ 𝐴 )
2	1	sscond	⊢ ( 𝑋 ∈ 𝐴 → ( 𝐵 ∖ 𝐴 ) ⊆ ( 𝐵 ∖ { 𝑋 } ) )
3	2	sselda	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝑌 ∈ ( 𝐵 ∖ { 𝑋 } ) )