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Theorem 0pss 3813
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 3774 . . 3  |-  (/)  C_  A
2 df-pss 3431 . . 3  |-  ( (/)  C.  A  <->  ( (/)  C_  A  /\  (/)  =/=  A ) )
31, 2mpbiran 934 . 2  |-  ( (/)  C.  A  <->  (/)  =/=  A )
4 necom 2688 . 2  |-  ( (/)  =/=  A  <->  A  =/=  (/) )
53, 4bitri 257 1  |-  ( (/)  C.  A  <->  A  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    =/= wne 2632    C_ wss 3415    C. wpss 3416   (/)c0 3742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743
This theorem is referenced by:  php  7781  zornn0g  8960  prn0  9439  genpn0  9453  nqpr  9464  ltexprlem5  9490  reclem2pr  9498  suplem1pr  9502  alexsubALTlem4  21113  bj-2upln0  31661  bj-2upln1upl  31662  0pssin  36409
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