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Theorem 0pss 4392
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 4347 . . 3 ∅ ⊆ 𝐴
2 df-pss 3951 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 705 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 3066 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 276 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wne 3013  wss 3933  wpss 3934  c0 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-dif 3936  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289
This theorem is referenced by:  php  8689  zornn0g  9915  prn0  10399  genpn0  10413  nqpr  10424  ltexprlem5  10450  reclem2pr  10458  suplem1pr  10462  alexsubALTlem4  22586  bj-2upln0  34232  bj-2upln1upl  34233  0pssin  39994
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