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Theorem npss0 3833
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 3793 . . . 4  |-  (/)  C_  A
21a1i 11 . . 3  |-  ( A 
C_  (/)  ->  (/)  C_  A
)
3 iman 425 . . 3  |-  ( ( A  C_  (/)  ->  (/)  C_  A
)  <->  -.  ( A  C_  (/)  /\  -.  (/)  C_  A
) )
42, 3mpbi 211 . 2  |-  -.  ( A  C_  (/)  /\  -.  (/)  C_  A
)
5 dfpss3 3551 . 2  |-  ( A 
C.  (/)  <->  ( A  C_  (/) 
/\  -.  (/)  C_  A
) )
64, 5mtbir 300 1  |-  -.  A  C.  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    C_ wss 3436    C. wpss 3437   (/)c0 3761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-v 3082  df-dif 3439  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762
This theorem is referenced by:  pssnn  7799
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