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Theorem 0pss 3492
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 3483 . . 3  |-  (/)  C_  A
2 df-pss 3168 . . 3  |-  ( (/)  C.  A  <->  ( (/)  C_  A  /\  (/)  =/=  A ) )
31, 2mpbiran 884 . 2  |-  ( (/)  C.  A  <->  (/)  =/=  A )
4 necom 2527 . 2  |-  ( (/)  =/=  A  <->  A  =/=  (/) )
53, 4bitri 240 1  |-  ( (/)  C.  A  <->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    =/= wne 2446    C_ wss 3152    C. wpss 3153   (/)c0 3455
This theorem is referenced by:  php  7045  zornn0g  8132  prn0  8613  genpn0  8627  nqpr  8638  ltexprlem5  8664  reclem2pr  8672  suplem1pr  8676  alexsubALTlem4  17744
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456
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