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Theorem pssdifn0 4326
Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)
Assertion
Ref Expression
pssdifn0 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)

Proof of Theorem pssdifn0
StepHypRef Expression
1 ssdif0 4324 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴) = ∅)
2 eqss 3960 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32simplbi2 502 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐴 = 𝐵))
41, 3biimtrrid 242 . . 3 (𝐴𝐵 → ((𝐵𝐴) = ∅ → 𝐴 = 𝐵))
54necon3d 2961 . 2 (𝐴𝐵 → (𝐴𝐵 → (𝐵𝐴) ≠ ∅))
65imp 408 1 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = 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:  pssdif  4327  tz7.7  6344  domdifsn  9001  inf3lem3  9571  isf32lem6  10299  fclscf  23392  flimfnfcls  23395  lebnumlem1  24340  lebnumlem2  24341  lebnumlem3  24342  ig1peu  25552  ig1pdvds  25557  qsidomlem2  32274  divrngidl  36533
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