Theorem elneeldif 3965

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

Assertion

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

Proof of Theorem elneeldif

Step	Hyp	Ref	Expression
1		eldif 3961	. . 3 ⊢ (𝑌 ∈ (𝐵 ∖ 𝐴) ↔ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴))
2		nelne2 3040	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌)
3	2	ex 412	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑌 ∈ 𝐴 → 𝑋 ≠ 𝑌))
4	3	adantld 490	. . 3 ⊢ (𝑋 ∈ 𝐴 → ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌))
5	1, 4	biimtrid 242	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑌 ∈ (𝐵 ∖ 𝐴) → 𝑋 ≠ 𝑌))
6	5	imp 406	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2108   ≠ wne 2940   ∖ cdif 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954

This theorem is referenced by:  frlmsslsp  21816  fmlasucdisj  35404  mhpind  42604