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Theorem elneeldif 3976
Description: The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023.)
Assertion
Ref Expression
elneeldif ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑋𝑌)

Proof of Theorem elneeldif
StepHypRef Expression
1 eldif 3972 . . 3 (𝑌 ∈ (𝐵𝐴) ↔ (𝑌𝐵 ∧ ¬ 𝑌𝐴))
2 nelne2 3037 . . . . 5 ((𝑋𝐴 ∧ ¬ 𝑌𝐴) → 𝑋𝑌)
32ex 412 . . . 4 (𝑋𝐴 → (¬ 𝑌𝐴𝑋𝑌))
43adantld 490 . . 3 (𝑋𝐴 → ((𝑌𝐵 ∧ ¬ 𝑌𝐴) → 𝑋𝑌))
51, 4biimtrid 242 . 2 (𝑋𝐴 → (𝑌 ∈ (𝐵𝐴) → 𝑋𝑌))
65imp 406 1 ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  wne 2937  cdif 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-v 3479  df-dif 3965
This theorem is referenced by:  frlmsslsp  21833  fmlasucdisj  35383  mhpind  42580
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