Theorem elneeldif 3913

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 3909	. . 3 ⊢ (𝑌 ∈ (𝐵 ∖ 𝐴) ↔ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴))
2		nelne2 3054	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌)
3	2	ex 418	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑌 ∈ 𝐴 → 𝑋 ≠ 𝑌))
4	3	adantld 496	. . 3 ⊢ (𝑋 ∈ 𝐴 → ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌))
5	1, 4	biimtrid 245	. 2 ⊢ (𝑋 ∈ 𝐴 → (𝑌 ∈ (𝐵 ∖ 𝐴) → 𝑋 ≠ 𝑌))
6	5	imp 412	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑋 ≠ 𝑌)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145   ≠ wne 2956   ∖ cdif 3896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3453  df-dif 3902

This theorem is used by:  frlmsslsp  22095  fmlasucdisj  36143  mh-inf3f1  37309  mhpind  43602