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Theorem pssdifn0 4142
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 4140 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴) = ∅)
2 eqss 3811 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32simplbi2 495 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐴 = 𝐵))
41, 3syl5bir 235 . . 3 (𝐴𝐵 → ((𝐵𝐴) = ∅ → 𝐴 = 𝐵))
54necon3d 2990 . 2 (𝐴𝐵 → (𝐴𝐵 → (𝐵𝐴) ≠ ∅))
65imp 396 1 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wne 2969  cdif 3764  wss 3767  c0 4113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-v 3385  df-dif 3770  df-in 3774  df-ss 3781  df-nul 4114
This theorem is referenced by:  pssdif  4143  tz7.7  5965  domdifsn  8283  inf3lem3  8775  isf32lem6  9466  fclscf  22153  flimfnfcls  22156  lebnumlem1  23084  lebnumlem2  23085  lebnumlem3  23086  ig1peu  24268  ig1pdvds  24273  divrngidl  34305
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