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Theorem 0pss 4379
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 4333 . . 3 ∅ ⊆ 𝐴
2 df-pss 3938 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 708 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 3067 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 278 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wne 3014  wss 3919  wpss 3920  c0 4276
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3015  df-v 3482  df-dif 3922  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277
This theorem is referenced by:  php  8694  zornn0g  9921  prn0  10405  genpn0  10419  nqpr  10430  ltexprlem5  10456  reclem2pr  10464  suplem1pr  10468  alexsubALTlem4  22653  bj-2upln0  34373  bj-2upln1upl  34374  0pssin  40328
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