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Theorem npss0 3659
Description: No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
npss0  |-  -.  A  C.  (/)

Proof of Theorem npss0
StepHypRef Expression
1 0ss 3649 . . . 4  |-  (/)  C_  A
21a1i 11 . . 3  |-  ( A 
C_  (/)  ->  (/)  C_  A
)
3 iman 414 . . 3  |-  ( ( A  C_  (/)  ->  (/)  C_  A
)  <->  -.  ( A  C_  (/)  /\  -.  (/)  C_  A
) )
42, 3mpbi 200 . 2  |-  -.  ( A  C_  (/)  /\  -.  (/)  C_  A
)
5 dfpss3 3426 . 2  |-  ( A 
C.  (/)  <->  ( A  C_  (/) 
/\  -.  (/)  C_  A
) )
64, 5mtbir 291 1  |-  -.  A  C.  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    C_ wss 3313    C. wpss 3314   (/)c0 3621
This theorem is referenced by:  pssnn  7320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2951  df-dif 3316  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622
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