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Theorem pssdifn0 3923
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 3921 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴) = ∅)
2 eqss 3602 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32simplbi2 654 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐴 = 𝐵))
41, 3syl5bir 233 . . 3 (𝐴𝐵 → ((𝐵𝐴) = ∅ → 𝐴 = 𝐵))
54necon3d 2811 . 2 (𝐴𝐵 → (𝐴𝐵 → (𝐵𝐴) ≠ ∅))
65imp 445 1 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wne 2790  cdif 3556  wss 3559  c0 3896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3191  df-dif 3562  df-in 3566  df-ss 3573  df-nul 3897
This theorem is referenced by:  pssdif  3924  tz7.7  5713  domdifsn  7995  inf3lem3  8479  isf32lem6  9132  fclscf  21752  flimfnfcls  21755  lebnumlem1  22683  lebnumlem2  22684  lebnumlem3  22685  ig1peu  23852  ig1pdvds  23857  divrngidl  33494
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