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Theorem 0pss 4175
Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.)
Assertion
Ref Expression
0pss (∅ ⊊ 𝐴𝐴 ≠ ∅)

Proof of Theorem 0pss
StepHypRef Expression
1 0ss 4134 . . 3 ∅ ⊆ 𝐴
2 df-pss 3748 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 700 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 2990 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 266 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wne 2937  wss 3732  wpss 3733  c0 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3735  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080
This theorem is referenced by:  php  8351  zornn0g  9580  prn0  10064  genpn0  10078  nqpr  10089  ltexprlem5  10115  reclem2pr  10123  suplem1pr  10127  alexsubALTlem4  22133  bj-2upln0  33438  bj-2upln1upl  33439  0pssin  38738
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