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Theorem npss0 4385
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 4336 . 2 ∅ ⊆ 𝐴
2 ssnpss 4043 . 2 (∅ ⊆ 𝐴 → ¬ 𝐴 ⊊ ∅)
31, 2ax-mp 5 1 ¬ 𝐴 ⊊ ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wss 3892  wpss 3893  c0 4262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-v 3433  df-dif 3895  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263
This theorem is referenced by:  pssnn  8925  pssnnOLD  9010  pssn0  40191
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