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Theorem difn0 4244
 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 3944 . . 3 (𝐴 = 𝐵𝐴𝐵)
2 ssdif0 4243 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = ∅)
31, 2sylib 219 . 2 (𝐴 = 𝐵 → (𝐴𝐵) = ∅)
43necon3i 3016 1 ((𝐴𝐵) ≠ ∅ → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1522   ≠ wne 2984   ∖ cdif 3856   ⊆ wss 3859  ∅c0 4211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-v 3439  df-dif 3862  df-in 3866  df-ss 3874  df-nul 4212 This theorem is referenced by:  disjdsct  30126  bj-2upln1upl  33941
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