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Theorem 0pss 3610
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 3601 . . 3  |-  (/)  C_  A
2 df-pss 3281 . . 3  |-  ( (/)  C.  A  <->  ( (/)  C_  A  /\  (/)  =/=  A ) )
31, 2mpbiran 885 . 2  |-  ( (/)  C.  A  <->  (/)  =/=  A )
4 necom 2633 . 2  |-  ( (/)  =/=  A  <->  A  =/=  (/) )
53, 4bitri 241 1  |-  ( (/)  C.  A  <->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    =/= wne 2552    C_ wss 3265    C. wpss 3266   (/)c0 3573
This theorem is referenced by:  php  7229  zornn0g  8320  prn0  8801  genpn0  8815  nqpr  8826  ltexprlem5  8852  reclem2pr  8860  suplem1pr  8864  alexsubALTlem4  18004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-v 2903  df-dif 3268  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574
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