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Theorem pssdifn0 4328
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 4326 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴) = ∅)
2 eqss 3985 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32simplbi2 501 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐴 = 𝐵))
41, 3syl5bir 244 . . 3 (𝐴𝐵 → ((𝐵𝐴) = ∅ → 𝐴 = 𝐵))
54necon3d 3041 . 2 (𝐴𝐵 → (𝐴𝐵 → (𝐵𝐴) ≠ ∅))
65imp 407 1 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wne 3020  cdif 3936  wss 3939  c0 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-v 3501  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295
This theorem is referenced by:  pssdif  4329  tz7.7  6214  domdifsn  8592  inf3lem3  9085  isf32lem6  9772  fclscf  22549  flimfnfcls  22552  lebnumlem1  23480  lebnumlem2  23481  lebnumlem3  23482  ig1peu  24680  ig1pdvds  24685  qsidomlem2  30871  divrngidl  35175
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