Theorem elneeldif

Description: The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023)

	Ref	Expression
Assertion	elneeldif	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝑋 ≠ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		eldif	⊢ ( 𝑌 ∈ ( 𝐵 ∖ 𝐴 ) ↔ ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴 ) )
2		nelne2	⊢ ( ( 𝑋 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐴 ) → 𝑋 ≠ 𝑌 )
3	2	ex	⊢ ( 𝑋 ∈ 𝐴 → ( ¬ 𝑌 ∈ 𝐴 → 𝑋 ≠ 𝑌 ) )
4	3	adantld	⊢ ( 𝑋 ∈ 𝐴 → ( ( 𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴 ) → 𝑋 ≠ 𝑌 ) )
5	1 4	biimtrid	⊢ ( 𝑋 ∈ 𝐴 → ( 𝑌 ∈ ( 𝐵 ∖ 𝐴 ) → 𝑋 ≠ 𝑌 ) )
6	5	imp	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝑋 ≠ 𝑌 )