Theorem elneeldif 3916

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 3912	. . 3 ⊢ (𝑌 ∈ (𝐵 ∖ 𝐴) ↔ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴))
2		nelne2 3026	. . . . 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 2111   ≠ wne 2928   ∖ cdif 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3905

This theorem is referenced by:  frlmsslsp  21734  fmlasucdisj  35441  mhpind  42633