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Theorem 0pss 3657
Description: The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
Assertion
Ref Expression
0pss  |-  ( (/)  C.  A  <->  A  =/=  (/) )

Proof of Theorem 0pss
StepHypRef Expression
1 0ss 3648 . . 3  |-  (/)  C_  A
2 df-pss 3328 . . 3  |-  ( (/)  C.  A  <->  ( (/)  C_  A  /\  (/)  =/=  A ) )
31, 2mpbiran 885 . 2  |-  ( (/)  C.  A  <->  (/)  =/=  A )
4 necom 2679 . 2  |-  ( (/)  =/=  A  <->  A  =/=  (/) )
53, 4bitri 241 1  |-  ( (/)  C.  A  <->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    =/= wne 2598    C_ wss 3312    C. wpss 3313   (/)c0 3620
This theorem is referenced by:  php  7283  zornn0g  8375  prn0  8856  genpn0  8870  nqpr  8881  ltexprlem5  8907  reclem2pr  8915  suplem1pr  8919  alexsubALTlem4  18071
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621
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