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Theorem 0pss 3827
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 3777 . . 3  |-  (/)  C_  A
2 df-pss 3455 . . 3  |-  ( (/)  C.  A  <->  ( (/)  C_  A  /\  (/)  =/=  A ) )
31, 2mpbiran 909 . 2  |-  ( (/)  C.  A  <->  (/)  =/=  A )
4 necom 2721 . 2  |-  ( (/)  =/=  A  <->  A  =/=  (/) )
53, 4bitri 249 1  |-  ( (/)  C.  A  <->  A  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    =/= wne 2648    C_ wss 3439    C. wpss 3440   (/)c0 3748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3442  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749
This theorem is referenced by:  php  7608  zornn0g  8788  prn0  9272  genpn0  9286  nqpr  9297  ltexprlem5  9323  reclem2pr  9331  suplem1pr  9335  alexsubALTlem4  19757  bj-2upln0  32868  bj-2upln1upl  32869
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