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Theorem difn0 3897
 Description: If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
Assertion
Ref Expression
difn0 ((𝐴𝐵) ≠ ∅ → 𝐴𝐵)

Proof of Theorem difn0
StepHypRef Expression
1 eqimss 3620 . . 3 (𝐴 = 𝐵𝐴𝐵)
2 ssdif0 3896 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
31, 2sylib 207 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = ∅)
43necon3i 2814 1 ((𝐴𝐵) ≠ ∅ → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ≠ wne 2780   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  disjdsct  28863  bj-2upln1upl  32205
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