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Theorem difn0 4142
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 3852 . . 3 (𝐴 = 𝐵𝐴𝐵)
2 ssdif0 4141 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
31, 2sylib 210 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = ∅)
43necon3i 3002 1 ((𝐴𝐵) ≠ ∅ → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wne 2970  cdif 3765  wss 3768  c0 4114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-v 3386  df-dif 3771  df-in 3775  df-ss 3782  df-nul 4115
This theorem is referenced by:  disjdsct  29991  bj-2upln1upl  33497
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