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Theorem pssdifn0 4265
 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 4263 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴) = ∅)
2 eqss 3908 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32simplbi2 505 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐴 = 𝐵))
41, 3syl5bir 246 . . 3 (𝐴𝐵 → ((𝐵𝐴) = ∅ → 𝐴 = 𝐵))
54necon3d 2973 . 2 (𝐴𝐵 → (𝐴𝐵 → (𝐵𝐴) ≠ ∅))
65imp 411 1 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ≠ wne 2952   ∖ cdif 3856   ⊆ wss 3859  ∅c0 4226 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-v 3412  df-dif 3862  df-in 3866  df-ss 3876  df-nul 4227 This theorem is referenced by:  pssdif  4266  tz7.7  6196  domdifsn  8621  inf3lem3  9119  isf32lem6  9811  fclscf  22718  flimfnfcls  22721  lebnumlem1  23655  lebnumlem2  23656  lebnumlem3  23657  ig1peu  24864  ig1pdvds  24869  qsidomlem2  31143  divrngidl  35739
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