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Theorem 0pss 4445
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 4397 . . 3 ∅ ⊆ 𝐴
2 df-pss 3968 . . 3 (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
31, 2mpbiran 708 . 2 (∅ ⊊ 𝐴 ↔ ∅ ≠ 𝐴)
4 necom 2995 . 2 (∅ ≠ 𝐴𝐴 ≠ ∅)
53, 4bitri 275 1 (∅ ⊊ 𝐴𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wne 2941  wss 3949  wpss 3950  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324
This theorem is referenced by:  php  9210  phpOLD  9222  zornn0g  10500  prn0  10984  genpn0  10998  nqpr  11009  ltexprlem5  11035  reclem2pr  11043  suplem1pr  11047  alexsubALTlem4  23554  bj-2upln0  35904  bj-2upln1upl  35905  0pssin  42522
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