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Theorem npss0 4450
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 4401 . 2 ∅ ⊆ 𝐴
2 ssnpss 4102 . 2 (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
31, 2ax-mp 5 1 ¬ 𝐴 ⊊ ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wss 3947  wpss 3948  c0 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-dif 3950  df-ss 3964  df-pss 3967  df-nul 4326
This theorem is referenced by:  pssnn  9206  pssnnOLD  9299  pssn0  41949
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