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Theorem difn0 4281
 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 3974 . . 3 (𝐴 = 𝐵𝐴𝐵)
2 ssdif0 4280 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
31, 2sylib 221 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = ∅)
43necon3i 3022 1 ((𝐴𝐵) ≠ ∅ → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ≠ wne 2990   ∖ cdif 3881   ⊆ wss 3884  ∅c0 4246 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247 This theorem is referenced by:  disjdsct  30466  bj-2upln1upl  34461
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