MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pssdifn0 Structured version   Visualization version   GIF version

Theorem pssdifn0 4052
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 4050 . . . 4 (𝐵𝐴 ↔ (𝐵𝐴) = ∅)
2 eqss 3724 . . . . 5 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
32simplbi2 656 . . . 4 (𝐴𝐵 → (𝐵𝐴𝐴 = 𝐵))
41, 3syl5bir 233 . . 3 (𝐴𝐵 → ((𝐵𝐴) = ∅ → 𝐴 = 𝐵))
54necon3d 2917 . 2 (𝐴𝐵 → (𝐴𝐵 → (𝐵𝐴) ≠ ∅))
65imp 444 1 ((𝐴𝐵𝐴𝐵) → (𝐵𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wne 2896  cdif 3677  wss 3680  c0 4023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694  df-nul 4024
This theorem is referenced by:  pssdif  4053  tz7.7  5862  domdifsn  8159  inf3lem3  8640  isf32lem6  9293  fclscf  21951  flimfnfcls  21954  lebnumlem1  22882  lebnumlem2  22883  lebnumlem3  22884  ig1peu  24051  ig1pdvds  24056  divrngidl  34059
  Copyright terms: Public domain W3C validator