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Theorem 0pss 4444
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 4396 . . 3 ∅ ⊆ 𝐴
2 df-pss 3967 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 707 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 2994 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 274 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wne 2940  wss 3948  wpss 3949  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323
This theorem is referenced by:  php  9212  phpOLD  9224  zornn0g  10502  prn0  10986  genpn0  11000  nqpr  11011  ltexprlem5  11037  reclem2pr  11045  suplem1pr  11049  alexsubALTlem4  23561  bj-2upln0  35996  bj-2upln1upl  35997  0pssin  42610
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