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Theorem 0pss 3852
Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)
Assertion
Ref Expression
0pss  |-  ( (/)  C.  A  <->  A  =/=  (/) )

Proof of Theorem 0pss
StepHypRef Expression
1 0ss 3813 . . 3  |-  (/)  C_  A
2 df-pss 3477 . . 3  |-  ( (/)  C.  A  <->  ( (/)  C_  A  /\  (/)  =/=  A ) )
31, 2mpbiran 916 . 2  |-  ( (/)  C.  A  <->  (/)  =/=  A )
4 necom 2723 . 2  |-  ( (/)  =/=  A  <->  A  =/=  (/) )
53, 4bitri 249 1  |-  ( (/)  C.  A  <->  A  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    =/= wne 2649    C_ wss 3461    C. wpss 3462   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784
This theorem is referenced by:  php  7694  zornn0g  8876  prn0  9356  genpn0  9370  nqpr  9381  ltexprlem5  9407  reclem2pr  9415  suplem1pr  9419  alexsubALTlem4  20716  bj-2upln0  34982  bj-2upln1upl  34983  0pssin  38244
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