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Theorem npss0 4379
Description: No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
Assertion
Ref Expression
npss0 ¬ 𝐴 ⊊ ∅

Proof of Theorem npss0
StepHypRef Expression
1 0ss 4330 . 2 ∅ ⊆ 𝐴
2 ssnpss 4038 . 2 (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
31, 2ax-mp 5 1 ¬ 𝐴 ⊊ ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wss 3887  wpss 3888  c0 4256
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257
This theorem is referenced by:  pssnn  8951  pssnnOLD  9040  pssn0  40202
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