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Metamath Proof Explorer

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 } ) )