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Theorem elneeldif 3855
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 3851 . . 3 (𝑌 ∈ (𝐵𝐴) ↔ (𝑌𝐵 ∧ ¬ 𝑌𝐴))
2 nelne2 3031 . . . . 5 ((𝑋𝐴 ∧ ¬ 𝑌𝐴) → 𝑋𝑌)
32ex 416 . . . 4 (𝑋𝐴 → (¬ 𝑌𝐴𝑋𝑌))
43adantld 494 . . 3 (𝑋𝐴 → ((𝑌𝐵 ∧ ¬ 𝑌𝐴) → 𝑋𝑌))
51, 4syl5bi 245 . 2 (𝑋𝐴 → (𝑌 ∈ (𝐵𝐴) → 𝑋𝑌))
65imp 410 1 ((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wcel 2113  wne 2934  cdif 3838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3399  df-dif 3844
This theorem is referenced by:  frlmsslsp  20605  fmlasucdisj  32924  mhpind  39846
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