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Theorem npss0 3609
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 3599 . . . 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 3376 . 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 3263    C. wpss 3264   (/)c0 3571
This theorem is referenced by:  pssnn  7263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572
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