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Theorem 0pss 3505
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 3496 . . 3  |-  (/)  C_  A
2 df-pss 3181 . . 3  |-  ( (/)  C.  A  <->  ( (/)  C_  A  /\  (/)  =/=  A ) )
31, 2mpbiran 884 . 2  |-  ( (/)  C.  A  <->  (/)  =/=  A )
4 necom 2540 . 2  |-  ( (/)  =/=  A  <->  A  =/=  (/) )
53, 4bitri 240 1  |-  ( (/)  C.  A  <->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    =/= wne 2459    C_ wss 3165    C. wpss 3166   (/)c0 3468
This theorem is referenced by:  php  7061  zornn0g  8148  prn0  8629  genpn0  8643  nqpr  8654  ltexprlem5  8680  reclem2pr  8688  suplem1pr  8692  alexsubALTlem4  17760
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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