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Theorem difn0 3896
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 3619 . . 3 (𝐴 = 𝐵𝐴𝐵)
2 ssdif0 3895 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
31, 2sylib 206 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = ∅)
43necon3i 2813 1 ((𝐴𝐵) ≠ ∅ → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wne 2779  cdif 3536  wss 3539  c0 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874
This theorem is referenced by:  disjdsct  28697  bj-2upln1upl  32029
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