Theorem elneeldif 3927

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 3923	. . 3 ⊢ (𝑌 ∈ (𝐵 ∖ 𝐴) ↔ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴))
2		nelne2 3039	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌)
3	2	ex 413	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑌 ∈ 𝐴 → 𝑋 ≠ 𝑌))
4	3	adantld 491	. . 3 ⊢ (𝑋 ∈ 𝐴 → ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌))
5	1, 4	biimtrid 241	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑌 ∈ (𝐵 ∖ 𝐴) → 𝑋 ≠ 𝑌))
6	5	imp 407	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2106   ≠ wne 2939   ∖ cdif 3910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3448  df-dif 3916

This theorem is referenced by:  frlmsslsp  21239  fmlasucdisj  34080  mhpind  40827