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Theorem difn0 4325
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 4001 . . 3 (𝐴 = 𝐵𝐴𝐵)
2 ssdif0 4324 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
31, 2sylib 217 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = ∅)
43necon3i 2973 1 ((𝐴𝐵) ≠ ∅ → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wne 2940  cdif 3908  wss 3911  c0 4283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284
This theorem is referenced by:  disjdsct  31663  bj-2upln1upl  35541
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