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Theorem npss0 3506
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 3496 . . . 4  |-  (/)  C_  A
21a1i 10 . . 3  |-  ( A 
C_  (/)  ->  (/)  C_  A
)
3 iman 413 . . 3  |-  ( ( A  C_  (/)  ->  (/)  C_  A
)  <->  -.  ( A  C_  (/)  /\  -.  (/)  C_  A
) )
42, 3mpbi 199 . 2  |-  -.  ( A  C_  (/)  /\  -.  (/)  C_  A
)
5 dfpss3 3275 . 2  |-  ( A 
C.  (/)  <->  ( A  C_  (/) 
/\  -.  (/)  C_  A
) )
64, 5mtbir 290 1  |-  -.  A  C.  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    C_ wss 3165    C. wpss 3166   (/)c0 3468
This theorem is referenced by:  pssnn  7097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469
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